Content check — Intermediate Algebra, Chapter 10: Exponential and Logarithmic Functions

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 10: Exponential and Logarithmic Functions

343 problems · 60 figures

Be Prepared 10.1 §10.1 source ↗

f (x) = 2 x - 3

and

g (x) =

x^{2} +

2 x -

3,

find

f (4) .

If you missed this problem, review Example 3.48.

Example 3.48 in the book ↗

Answer

ⓐ

f 4 = 5

; ⓑ

g f 4 = 32

Be Prepared 10.2 §10.1 source ↗

Solve for

x,

3 x + 2 y = 12 .

If you missed this problem, review Example 2.31.

Example 2.31 in the book ↗

Answer

x = - \frac{2}{3} y + 4

Be Prepared 10.3 §10.1 source ↗

Simplify:

5 \frac{(x + 4)}{5} - 4 .

If you missed this problem, review Example 1.25.

Example 1.25 in the book ↗

Answer

x

Try It 10.1 §10.1 source ↗

For functions

f (x) = 3 x - 2

and

g (x) = 5 x + 1,

find ⓐ

(f \circ g) (x)

ⓑ

(g \circ f) (x)

ⓒ

(f \cdot g) (x)

Answer

ⓐ

15 x + 1

ⓑ

15 x - 9

ⓒ

15 x^{2} - 7 x - 2

Try It 10.2 §10.1 source ↗

For functions

f (x) = 4 x - 3,

and

g (x) = 6 x - 5,

find ⓐ

(f \circ g) (x),

ⓑ

(g \circ f) (x),

and ⓒ

(f \cdot g) (x) .

Answer

ⓐ

24 x - 23

ⓑ

24 x - 23

ⓒ

24 x^{2} - 38 x + 15

Try It 10.3 §10.1 source ↗

For functions

f (x) = x^{2} - 9,

and

g (x) = 2 x + 5,

find ⓐ

(f \circ g) (−2),

ⓑ

(g \circ f) (−3),

and ⓒ

(f \circ f) (4) .

Answer

ⓐ –8 ⓑ 5 ⓒ 40

Try It 10.4 §10.1 source ↗

For functions

f (x) = x^{2} + 1,

and

g (x) = 3 x - 5,

find ⓐ

(f \circ g) (−1),

ⓑ

(g \circ f) (2),

and ⓒ

(f \circ f) (−1) .

Answer

ⓐ 65 ⓑ 10 ⓒ 5

Try It 10.5 §10.1 source ↗

For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

ⓐ ${(−3, −6), (−2, −4), (−1, −2), (0, 0), (1, 2), (2, 4), (3, 6)}$ ⓑ ${(−4, 8), (−2, 4), (−1, 2), (0, 0), (1, 2), (2, 4), (4, 8)}$

Answer

ⓐ One-to-one function
ⓑ Function; not one-to-one

Try It 10.6 §10.1 source ↗

For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

ⓐ ${(27, −3), (8, −2), (1, −1), (0, 0), (1, 1), (8, 2), (27, 3)}$ ⓑ ${(7, −3), (−5, −4), (8, 0), (0, 0), (−6, 4), (−2, 2), (−1, 3)}$

Answer

ⓐ Not a function
ⓑ Function; not one-to-one

Try It 10.7 §10.1 source ↗

Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.

Graph a shows a parabola opening to the right with vertex at (negative 1, 0). Graph b shows an exponential function that does not cross the x axis and that passes through (0, 1) before increasing rapidly.

Answer

ⓐ Not a function ⓑ One-to-one function

Try It 10.8 §10.1 source ↗

Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.

Graph a shows a parabola opening up with vertex at (0, 3). Graph b shows a straight line passing through (0, negative 2) and (2, 0).

Answer

ⓐ Function; not one-to-one ⓑ One-to-one function

Try It 10.9 §10.1 source ↗

Find the inverse of

{(0, 4), (1, 7), (2, 10), (3, 13)} .

Determine the domain and range of the inverse function.

Answer

Inverse function:

{(4, 0), (7, 1), (10, 2), (13, 3)} .

Domain:

{4, 7, 10, 13} .

Range:

{0, 1, 2, 3} .

Try It 10.10 §10.1 source ↗

Find the inverse of

{(−1, 4), (−2, 1), (−3, 0), (−4, 2)} .

Determine the domain and range of the inverse function.

Answer

Inverse function:

{(4, −1), (1, −2), (0, −3), (2, −4)} .

Domain:

{0, 1, 2, 4} .

Range:

{−4, −3, −2, −1} .

Try It 10.11 §10.1 source ↗

Graph, on the same coordinate system, the inverse of the one-to one function.

The graph shows a line from (negative 3, negative 4) to (negative 2, negative 2) then to (0, negative 1), then to (1, 2) and then to (4, 3). The graph shows a line from (negative 3, 4) to (0, 3) then to (1, 2) and then to (4, 1).

Answer

This figure shows a line from (negative 4, negative 3) to (negative 2, negative 2) then to (negative 1, 0) then to (2, 1) and then to (3, 4).

Try It 10.12 §10.1 source ↗

Graph, on the same coordinate system, the inverse of the one-to one function.

A graph on a Cartesian coordinate plane shows a piecewise linear function connecting four points: (-3, 4), (0, 3), (1, 2), and (4, 1).

Answer

Graph extends from negative 4 to 4 on both axes. Points plotted are (negative 3, 4), (0, 3), (1, 2), and (4, 1). Line segments connect points.

Try It 10.13 §10.1 source ↗

Verify that the functions are inverse functions.

$f (x) = 4 x - 3$ and $g (x) = \frac{x + 3}{4} .$

Answer

g (f (x)) = x,

and

f (g (x)) = x,

so they are inverses.

Try It 10.14 §10.1 source ↗

Verify that the functions are inverse functions.

$f (x) = 2 x + 6$ and $g (x) = \frac{x - 6}{2} .$

Answer

g (f (x)) = x,

and

f (g (x)) = x,

so they are inverses.

Try It 10.15 §10.1 source ↗

Find the inverse of the function

f (x) = 5 x - 3 .

Answer

f^{−1} (x) = \frac{x + 3}{5}

Try It 10.16 §10.1 source ↗

Find the inverse of the function

f (x) = 8 x + 5 .

Answer

f^{−1} (x) = \frac{x - 5}{8}

Try It 10.17 §10.1 source ↗

Find the inverse of the function

f (x) = \sqrt[5]{3 x - 2} .

Answer

f^{−1} (x) =

\frac{x^{5} + 2}{3}

Try It 10.18 §10.1 source ↗

Find the inverse of the function

f (x) = \sqrt[4]{6 x - 7} .

Answer

f^{−1} (x) =

\frac{x^{4} + 7}{6}

10.1-1 §10.1 source ↗

Find and Evaluate Composite Functions

In the following exercises, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

$f (x) = 4 x + 3$ and $g (x) = 2 x + 5$

Answer

ⓐ

8 x + 23

ⓑ

8 x + 11

ⓒ

8 x^{2} + 26 x + 15

10.1-3 §10.1 source ↗

Find and Evaluate Composite Functions

In the following exercises, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

$f (x) = 6 x - 5$ and $g (x) = 4 x + 1$

Answer

ⓐ

24 x + 1

ⓑ

24 x - 19

ⓒ

24 x^{2} - 14 x - 5

10.1-5 §10.1 source ↗

Find and Evaluate Composite Functions

In the following exercises, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

$f (x) = 3 x$ and $g (x) = 2 x^{2} - 3 x$

Answer

ⓐ

6 x^{2} - 9 x

ⓑ

18 x^{2} - 9 x

ⓒ

6 x^{3} - 9 x^{2}

10.1-7 §10.1 source ↗

Find and Evaluate Composite Functions

In the following exercises, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

$f (x) = 2 x - 1$ and $g (x) = x^{2} + 2$

Answer

ⓐ

2 x^{2} + 3

ⓑ

4 x^{2} - 4 x + 3

ⓒ

2 x^{3} - x^{2} + 4 x - 2

10.1-9 §10.1 source ↗

In the following exercises, find the values described.

For functions $f (x) = 2 x^{2} + 3$ and $g (x) = 5 x - 1,$ find ⓐ $(f \circ g) (−2)$ ⓑ $(g \circ f) (−3)$ ⓒ $(f \circ f) (−1)$

Answer

ⓐ 245 ⓑ 104 ⓒ 53

10.1-11 §10.1 source ↗

In the following exercises, find the values described.

For functions $f (x) = 2 x^{3}$ and $g (x) = 3 x^{2} + 2,$ find ⓐ $(f \circ g) (−1)$ ⓑ $(g \circ f) (1)$ ⓒ $(g \circ g) (1)$

Answer

ⓐ 250 ⓑ 14 ⓒ 77

10.1-13 §10.1 source ↗

Determine Whether a Function is One-to-One

In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.

${(−3, 9), (−2, 4), (−1, 1), (0, 0)$ ,
$(1, 1),$ $(2, 4),$ $(3, 9)}$

Answer

Function; not one-to-one

10.1-15 §10.1 source ↗

Determine Whether a Function is One-to-One

In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.

${(−3, −5), (−2, −3), (−1, −1)$ ,
$(0, 1),$ $(1, 3),$ $(2, 5),$ $(3, 7)}$

Answer

One-to-one function

10.1-17 §10.1 source ↗

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

ⓐ

This figure shows a graph of a circle with center at the origin and radius 3.

ⓑ

This figure shows a graph of a parabola opening upward with vertex at (0k, 2).

Answer

ⓐ Not a function ⓑ Function; not one-to-one

10.1-19 §10.1 source ↗

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

ⓐ

This figure shows a graph of a curve that starts at (negative 6 negative 2) increases to the origin and then continues increasing slowly to (6, 2).

ⓑ

This figure shows a parabola opening upward with vertex at (0, negative 4).

Answer

ⓐ One-to-one function
ⓑ Function; not one-to-one

10.1-21 §10.1 source ↗

In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.

${(2, 1), (4, 2), (6, 3), (8, 4)}$

Answer

Inverse function:

{(1, 2), (2, 4), (3, 6), (4, 8)} .

Domain:

{1, 2, 3, 4} .

Range:

{2, 4, 6, 8} .

10.1-23 §10.1 source ↗

In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.

${(0, −2), (1, 3), (2, 7), (3, 12)}$

Answer

Inverse function:

{(−2, 0), (3, 1), (7, 2), (12, 3)} .

Domain:

{−2, 3, 7, 12} .

Range:

{0, 1, 2, 3} .

10.1-25 §10.1 source ↗

In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.

${(−2, −3), (−1, −1), (0, 1), (1, 3)}$

Answer

Inverse function:

{(−3, − 2), (−1, −1), (1, 0), (3, 1)} .

Domain:

{−3, − 1, 1, 3} .

Range:

{−2, −1, 0, 1} .

10.1-27 §10.1 source ↗

In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.

This figure shows a series of line segments from (negative 4, negative 3) to (negative 3, 0) then to (negative 1, 2) and then to (3, 4).

Answer

This figure shows a series of line segments from (negative 3, negative 4) to (0, negative 3) then to (2, negative 1), and then to (4, 3).

10.1-29 §10.1 source ↗

In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.

This figure shows a series of line segments from (negative 4, 4) to (0, 3) then to (3, 2) and then to (4, negative 1).

Answer

This figure shows a series of line segments from (negative 1, 4) to (2, 3) then to (3, 0), and then to (4, negative 4).

10.1-31 §10.1 source ↗

In the following exercises, determine whether or not the given functions are inverses.

$f (x) = x + 8$ and $g (x) = x - 8$

Answer

g (f (x)) = x,

and

f (g (x)) = x,

so they are inverses.

10.1-33 §10.1 source ↗

In the following exercises, determine whether or not the given functions are inverses.

$f (x) = 7 x$ and $g (x) = \frac{x}{7}$

Answer

g (f (x)) = x,

and

f (g (x)) = x,

so they are inverses.

10.1-35 §10.1 source ↗

In the following exercises, determine whether or not the given functions are inverses.

$f (x) = 7 x + 3$ and $g (x) = \frac{x - 3}{7}$

Answer

g (f (x)) = x,

and

f (g (x)) = x,

so they are inverses.

10.1-37 §10.1 source ↗

In the following exercises, determine whether or not the given functions are inverses.

$f (x) = \sqrt{x + 2}$ and $g (x) =$ $x^{2} -$ $2 (x > 0)$

Answer

g (f (x)) = x,

and

f (g (x)) = x,

so they are inverses (for nonnegative

x) .

10.1-39 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = x - 12$

Answer

f^{−1} (x) = x + 12

10.1-41 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = 9 x$

Answer

f^{−1} (x) = \frac{x}{9}

10.1-43 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = \frac{x}{6}$

Answer

f^{−1} (x) = 6 x

10.1-45 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = 6 x - 7$

Answer

f^{−1} (x) = \frac{x + 7}{6}

10.1-47 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = −2 x + 5$

Answer

f^{−1} (x) =

\frac{x - 5}{−2}

10.1-49 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = x^{2} + 6,$ $x \geq 0$

Answer

f^{−1} (x) = \sqrt{x - 6}

10.1-51 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = x^{3} - 4$

Answer

f^{−1} (x) = \sqrt[3]{x + 4}

10.1-53 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = \frac{1}{x + 2}$

Answer

f^{−1} (x) = \frac{1}{x} - 2

10.1-55 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = \sqrt{x - 2},$ $x \geq 2$

Answer

f^{−1} (x) = x^{2} + 2

x \geq 0

10.1-57 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = \sqrt[3]{x - 3}$

Answer

f^{−1} (x) = x^{3} + 3

10.1-59 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = \sqrt[4]{9 x - 5},$ $x \geq \frac{5}{9}$

Answer

f^{−1} (x) =

\frac{x^{4} + 5}{9}

x \geq 0

10.1-61 §10.1 source ↗

In the following exercises, find the inverse of each function.

$f (x) = \sqrt[5]{−3 x + 5}$

Answer

f^{−1} (x) =

\frac{x^{5} - 5}{−3}

10.1-63 §10.1 source ↗

In the following exercises, find the inverse of each function.

Explain how the graph of the inverse of a function is related to the graph of the function.

Answer

Answers will vary.

Be Prepared 10.4 §10.2 source ↗

Simplify:

(\frac{x^{3}}{x^{2}}) .

If you missed this problem, review Example 5.13.

Example 5.13 in the book ↗

Answer

x

Be Prepared 10.5 §10.2 source ↗

Evaluate: ⓐ

2^{0}

ⓑ

{(\frac{1}{3})}^{0} .

If you missed this problem, review Example 5.14.

Example 5.14 in the book ↗

Answer

ⓐ

1

; ⓑ

1

Be Prepared 10.6 §10.2 source ↗

Evaluate: ⓐ

2^{−1}

ⓑ

{(\frac{1}{3})}^{−1} .

If you missed this problem, review Example 5.15.

Example 5.15 in the book ↗

Answer

ⓐ

\frac{1}{2}

; ⓑ

3

Try It 10.19 §10.2 source ↗

Graph:

f (x) = 4^{x} .

Answer

This figure shows a curve that slopes swiftly upward from just above (negative 3, 0) through (0, 1) up to (1, 4).

Try It 10.20 §10.2 source ↗

Graph:

g (x) = 5^{x} .

Answer

This figure shows a curve that slopes swiftly upward from just above (negative 3, 0) through (0, 1) up to (1, 5).

Try It 10.21 §10.2 source ↗

Graph:

f (x) = {(\frac{1}{4})}^{x} .

Answer

This figure shows a curve that passes through (negative 1, 4), (0, 1) to a point just above (3, 0).

Try It 10.22 §10.2 source ↗

Graph:

g (x) = {(\frac{1}{5})}^{x} .

Answer

This figure shows a curve that passes through (negative 1, 5), (0, 1) to a point just above (3, 0).

Try It 10.23 §10.2 source ↗

On the same coordinate system, graph:

f (x) = 2^{x}

and

g (x) = 2^{x - 1} .

Answer

This figure shows the graphs of two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1) and (1, 2). The second function g of x equals 2 to the x minus 1 power is marked in red and passes through the points (0, 1 over 2), (1, 1), and (2, 2).

Try It 10.24 §10.2 source ↗

On the same coordinate system, graph:

f (x) = 3^{x}

and

g (x) = 3^{x + 1} .

Answer

This figure shows the graphs of two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1) and (1, 3). The second function g of x equals 3 to the x plus 1 power is marked in red and passes through the points (negative 2, 1 over 3), (negative 1, 1), and (0, 3).

Try It 10.25 §10.2 source ↗

On the same coordinate system, graph:

f (x) = 3^{x}

and

g (x) = 3^{x} + 2 .

Answer

Try It 10.26 §10.2 source ↗

On the same coordinate system, graph:

f (x) = 4^{x}

and

g (x) = 4^{x} - 2 .

Answer

This figure shows the graphs of two functions. The first function f of x equals 4 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 4), (0, 1) and (1, 4). The second function g of x equals 4 to the x power minus 2 is marked in red and passes through the points (negative 1, negative 7 over 4), (0, negative 1), and (1, 2).

Try It 10.27 §10.2 source ↗

Solve:

3^{3 x - 2} = 81 .

Answer

x = 2

Try It 10.28 §10.2 source ↗

Solve:

7^{x - 3} = 7 .

Answer

x = 4

Try It 10.29 §10.2 source ↗

Solve:

\frac{e^{x^{2}}}{e^{x}} = e^{2} .

Answer

x = −1, x = 2

Try It 10.30 §10.2 source ↗

Solve:

\frac{e^{x^{2}}}{e^{x}} = e^{6} .

Answer

x = −2, x = 3

Try It 10.31 §10.2 source ↗

Angela invested $$ 15,000$ in a savings account. If the interest rate is $4 %,$ how much will be in the account in 10 years by each method of compounding?

ⓐ compound quarterly

ⓑ compound monthly

Answer

ⓐ

$ 22,332.96

ⓑ

$ 22,362.49

ⓒ

$ 22,377.37

Try It 10.32 §10.2 source ↗

Allan invested $10,000 in a mutual fund. If the interest rate is $5 %,$ how much will be in the account in 15 years by each method of compounding?

ⓐ compound quarterly

ⓑ compound monthly

Answer

ⓐ $21,071.81 ⓑ $21,137.04
ⓒ $21,170.00

Try It 10.33 §10.2 source ↗

Another researcher at the Center for Disease Control and Prevention, Lisa, is studying the growth of a bacteria. She starts her experiment with 50 of the bacteria that grows at a rate of

15 %

per hour. She will check on the bacteria every 8 hours. How many bacteria will she find in 8 hours?

Answer

She will find 166 bacteria.

Try It 10.34 §10.2 source ↗

Milan, a biologist is observing the growth pattern of a virus. They start with 100 of the virus that grows at a rate of

10 %

per hour. They will check on the virus in 24 hours. How many viruses will they find?

Answer

They will find 1,102 viruses.

10.2-65 §10.2 source ↗

Graph Exponential Functions

In the following exercises, graph each exponential function.

$f (x) = 2^{x}$

Answer

This figure shows a curve that passes through (negative 1, 1 over 2) through (0, 1) to (1, 2).

10.2-67 §10.2 source ↗

Graph Exponential Functions

In the following exercises, graph each exponential function.

$f (x) = 6^{x}$

Answer

This figure shows a curve that passes through (negative 1, 1 over 6) through (0, 1) to (1, 6).

10.2-69 §10.2 source ↗

Graph Exponential Functions

In the following exercises, graph each exponential function.

$f (x) = {(1.5)}^{x}$

Answer

This figure shows a curve that passes through (negative 1, 2 over 3) through (0, 1) to (1, 3 over 2).

10.2-71 §10.2 source ↗

Graph Exponential Functions

In the following exercises, graph each exponential function.

$f (x) = {(\frac{1}{2})}^{x}$

Answer

This figure shows a curve that passes through (negative 1, 2) through (0, 1) to (1, 1 over 2).

10.2-73 §10.2 source ↗

Graph Exponential Functions

In the following exercises, graph each exponential function.

$f (x) = {(\frac{1}{6})}^{x}$

Answer

This figure shows a curve that passes through (negative1, 6) through (0, 1) to (1, 1 over 6).

10.2-75 §10.2 source ↗

Graph Exponential Functions

In the following exercises, graph each exponential function.

$f (x) = {(0.4)}^{x}$

Answer

This figure shows a curve that passes through (negative 1, 5 over 2) through (0, 1) to (1, 2 over 5).

10.2-77 §10.2 source ↗

In the following exercises, graph each function in the same coordinate system.

$f (x) =$ $4^{x},$ $g (x) =$ $4^{x - 1}$

Answer

This figure shows two functions. The first function f of x equals 4 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 4), (0, 1) and (1, 4). The second function g of x equals 4 to the x minus 1 power is marked in red and passes through the points (0, 1 over 4), (1, 1) and (2, 4).

10.2-79 §10.2 source ↗

In the following exercises, graph each function in the same coordinate system.

$f (x) =$ $2^{x},$ $g (x) =$ $2^{x - 2}$

Answer

This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1) and (1, 2). The second function g of x equals 2 to the x minus 2 power is marked in red and passes through the points (0, 1 over 4), (1, 1 over 2), and (2, 1).

10.2-81 §10.2 source ↗

In the following exercises, graph each function in the same coordinate system.

$f (x) =$ $3^{x},$ $g (x) =$ $3^{x} +$ $2$

Answer

This figure shows two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1), and (1, 3). The second function g of x equals 3 to the x power plus 2 is marked in red and passes through the points (negative 2, 1), (negative 1, 3), and (0, 5).

10.2-83 §10.2 source ↗

In the following exercises, graph each function in the same coordinate system.

$f (x) =$ $2^{x},$ $g (x) =$ $2^{x} +$ $1$

Answer

10.2-85 §10.2 source ↗

In the following exercises, graph each exponential function.

$f (x) = 3^{x + 2}$

Answer

This figure shows an exponential curve that passes through (negative 3, 1 over 3), (negative 2, 1), and (0, 9).

10.2-87 §10.2 source ↗

In the following exercises, graph each exponential function.

$f (x) = 2^{x} + 3$

Answer

This figure shows an exponential that passes through (negative 1, 7 over 2), (0, 4), and (1, 5).

10.2-89 §10.2 source ↗

In the following exercises, graph each exponential function.

$f (x) =$ ${(\frac{1}{2})}^{x - 4}$

Answer

This figure shows an exponential that passes through (2, 4), (3, 2), and (4, 1).

10.2-91 §10.2 source ↗

In the following exercises, graph each exponential function.

$f (x) = e^{x} + 1$

Answer

This figure shows an exponential that passes through (1, 1 plus 1 over e), (0, 2), and (1, e).

10.2-93 §10.2 source ↗

In the following exercises, graph each exponential function.

$f (x) = − 2^{x}$

Answer

This figure shows an exponential that passes through (negative 1, negative 1 over 2), (0, negative 1), and (1, 2).

10.2-95 §10.2 source ↗

Solve Exponential Equations

In the following exercises, solve each equation.

$2^{3 x - 8} = 16$

Answer

x = 4

10.2-97 §10.2 source ↗

Solve Exponential Equations

In the following exercises, solve each equation.

$3^{x + 3} = 9$

Answer

x = −1

10.2-99 §10.2 source ↗

Solve Exponential Equations

In the following exercises, solve each equation.

$4^{x^{2}} = 4$

Answer

x = −1, x = 1

10.2-101 §10.2 source ↗

Solve Exponential Equations

In the following exercises, solve each equation.

$4^{x + 2} = 64$

Answer

x = 1

10.2-103 §10.2 source ↗

Solve Exponential Equations

In the following exercises, solve each equation.

$2^{x^{2} + 2 x} = \frac{1}{2}$

Answer

x = −1

10.2-105 §10.2 source ↗

Solve Exponential Equations

In the following exercises, solve each equation.

$e^{3 x} \cdot e^{4} = e^{10}$

Answer

x = 2

10.2-107 §10.2 source ↗

Solve Exponential Equations

In the following exercises, solve each equation.

$\frac{e^{x^{2}}}{e^{2}} = e^{x}$

Answer

x = −1, x = 2

10.2-109 §10.2 source ↗

In the following exercises, match the graphs to one of the following functions: ⓐ $2^{x}$ ⓑ $2^{x + 1}$ ⓒ $2^{x - 1}$ ⓓ $2^{x} + 2$ ⓔ $2^{x} - 2$ ⓕ $3^{x}$

This figure shows an exponential that passes through (1, 1 over 3), (0, 1), and (1, 3).

Answer

ⓕ

10.2-111 §10.2 source ↗

In the following exercises, match the graphs to one of the following functions: ⓐ $2^{x}$ ⓑ $2^{x + 1}$ ⓒ $2^{x - 1}$ ⓓ $2^{x} + 2$ ⓔ $2^{x} - 2$ ⓕ $3^{x}$

This figure shows an exponential that passes through (1, 1 over 2), (0, 1), and (1, 2).

Answer

ⓐ

10.2-113 §10.2 source ↗

In the following exercises, match the graphs to one of the following functions: ⓐ $2^{x}$ ⓑ $2^{x + 1}$ ⓒ $2^{x - 1}$ ⓓ $2^{x} + 2$ ⓔ $2^{x} - 2$ ⓕ $3^{x}$

This figure shows an exponential that passes through (negative 1, 3 over 2), (0, negative 1), and (1, 0).

Answer

ⓔ

10.2-115 §10.2 source ↗

Use exponential models in applications

In the following exercises, use an exponential model to solve.

Edgar accumulated $$ 5,000$ in credit card debt. If the interest rate is $20 %$ per year, and he does not make any payments for 2 years, how much will he owe on this debt in 2 years by each method of compounding? ⓐ compound quarterly ⓑ compound monthly ⓒ compound continuously

Answer

ⓐ

$ 7,387.28

ⓑ

$ 7,434.57

ⓒ

$ 7,459.12

10.2-117 §10.2 source ↗

Use exponential models in applications

In the following exercises, use an exponential model to solve.

Rochelle deposits $$ 5,000$ in an IRA. What will be the value of her investment in 25 years if the investment is earning $8 %$ per year and is compounded continuously?

Answer

$ 36,945.28

10.2-119 §10.2 source ↗

Use exponential models in applications

In the following exercises, use an exponential model to solve.

A researcher at the Center for Disease Control and Prevention is studying the growth of a bacteria. He starts his experiment with 100 of the bacteria that grows at a rate of $6 %$ per hour. He will check on the bacteria every 8 hours. How many bacteria will he find in 8 hours?

Answer

162 bacteria

10.2-121 §10.2 source ↗

Use exponential models in applications

In the following exercises, use an exponential model to solve.

In the last ten years the population of Indonesia has grown at a rate of $1.12 %$ per year to 258,316,051. If this rate continues, what will be the population in 10 more years?

Answer

288,929,825

10.2-123 §10.2 source ↗

Use exponential models in applications

In the following exercises, use an exponential model to solve.

Explain how you can distinguish between exponential functions and polynomial functions.

Answer

Answers will vary.

10.2-125 §10.2 source ↗

Use exponential models in applications

In the following exercises, use an exponential model to solve.

What happens to an exponential function as the values of $x$ decreases? Will the graph ever cross the
$x$ -axis? Explain.

Answer

Answers will vary.

Be Prepared 10.7 §10.3 source ↗

Solve:

x^{2} = 81 .

If you missed this problem, review Example 6.46.

Example 6.46 in the book ↗

Answer

x = 9, x = - 9

Be Prepared 10.8 §10.3 source ↗

Evaluate:

3^{−2} .

If you missed this problem, review Example 5.15.

Example 5.15 in the book ↗

Answer

\frac{1}{9}

Be Prepared 10.9 §10.3 source ↗

Solve:

2^{4} = 3 x - 5 .

If you missed this problem, review Example 2.2.

Example 2.2 in the book ↗

Answer

x = 7

Try It 10.35 §10.3 source ↗

Convert to logarithmic form: ⓐ

3^{2} = 9

ⓑ

7^{\frac{1}{2}} = \sqrt{7}

ⓒ

{(\frac{1}{3})}^{x} = \frac{1}{27}

Answer

ⓐ

{log}_{3} 9 = 2

ⓑ

{log}_{7} \sqrt{7} = \frac{1}{2}

ⓒ

{log}_{\frac{1}{3}} \frac{1}{27} = x

Try It 10.36 §10.3 source ↗

Convert to logarithmic form: ⓐ

4^{3} = 64

ⓑ

4^{\frac{1}{3}} = \sqrt[3]{4}

ⓒ

{(\frac{1}{2})}^{x} = \frac{1}{32}

Answer

ⓐ

{log}_{4} 64 = 3

ⓑ

{log}_{4} \sqrt[3]{4} = \frac{1}{3}

ⓒ

{log}_{\frac{1}{2}} \frac{1}{32} = x

Try It 10.37 §10.3 source ↗

Convert to exponential form: ⓐ

3 = {log}_{4} 64

ⓑ

0 = {log}_{x} 1

ⓒ

−2 =

{log}_{10} \frac{1}{100}

Answer

ⓐ

64 = 4^{3}

ⓑ

1 = x^{0}

ⓒ

\frac{1}{100} = 10^{−2}

Try It 10.38 §10.3 source ↗

Convert to exponential form: ⓐ

3 = {log}_{3} 27

ⓑ

0 = {log}_{3} 1

ⓒ

−1 = {log}_{10} \frac{1}{10}

Answer

ⓐ

27 = 3^{3}

ⓑ

1 = 3^{0}

ⓒ

\frac{1}{10} = 10^{−1}

Try It 10.39 §10.3 source ↗

Find the value of

x :

ⓐ

{log}_{x} 64 = 2

ⓑ

{log}_{5} x = 3

ⓒ

{log}_{\frac{1}{2}} \frac{1}{4} = x

Answer

ⓐ

x = 8

ⓑ

x = 125

ⓒ

x = 2

Try It 10.40 §10.3 source ↗

Find the value of

x :

ⓐ

{log}_{x} 81 = 2

ⓑ

{log}_{3} x = 5

ⓒ

{log}_{\frac{1}{3}} \frac{1}{27} = x

Answer

ⓐ

x = 9

ⓑ

x = 243

ⓒ

x = 3

Try It 10.41 §10.3 source ↗

Find the exact value of each logarithm without using a calculator: ⓐ

{log}_{12} 144

ⓑ

{log}_{4} 2

ⓒ

{log}_{2} \frac{1}{32}

Answer

ⓐ
2 ⓑ

\frac{1}{2}

ⓒ

−5

Try It 10.42 §10.3 source ↗

Find the exact value of each logarithm without using a calculator: ⓐ

{log}_{9} 81

ⓑ

{log}_{8} 2

ⓒ

{log}_{3} \frac{1}{9}

Answer

ⓐ 2 ⓑ

\frac{1}{3}

ⓒ

−2

Try It 10.43 §10.3 source ↗

Graph:

y = {log}_{3} x .

Answer

This figure shows the logarithmic curve going through the points (1 over 3, negative 1), (1, 0), and (3, 1).

Try It 10.44 §10.3 source ↗

Graph:

y = {log}_{5} x .

Answer

This figure shows the logarithmic curve going through the points (1 over 5, negative 1), (1, 0), and (5, 1).

Try It 10.45 §10.3 source ↗

Graph:

y = {log}_{\frac{1}{2}} x .

Answer

This figure shows the logarithmic curve going through the points (1 over 2, 1), (1, 0), and (2, negative 1).

Try It 10.46 §10.3 source ↗

Graph:

y = {log}_{\frac{1}{4}} x .

Answer

This figure shows the logarithmic curve going through the points (1 over 4, 1), (1, 0), and (4, negative 1).

Try It 10.47 §10.3 source ↗

Solve: ⓐ

{log}_{a} 121 = 2

ⓑ

ln x = 7

Answer

ⓐ

a = 11

ⓑ

x = e^{7}

Try It 10.48 §10.3 source ↗

Solve: ⓐ

{log}_{a} 64 = 3

ⓑ

ln x = 9

Answer

ⓐ

a = 4

ⓑ

x = e^{9}

Try It 10.49 §10.3 source ↗

Solve: ⓐ

{log}_{2} (5 x - 1) =

6

ⓑ

ln e^{3 x} = 6

Answer

ⓐ

x = 13

ⓑ

x = 2

Try It 10.50 §10.3 source ↗

Solve: ⓐ

{log}_{3} (4 x + 3) =

3

ⓑ

ln e^{4 x} = 4

Answer

ⓐ

x = 6

ⓑ

x = 1

Try It 10.51 §10.3 source ↗

What is the decibel level of one of the new quiet dishwashers with intensity

10^{−7}

watts per square inch?

Answer

The quiet dishwashers have a decibel level of 50 dB.

Try It 10.52 §10.3 source ↗

What is the decibel level heavy city traffic with intensity

10^{−3}

watts per square inch?

Answer

The decibel level of heavy traffic is 90 dB.

Try It 10.53 §10.3 source ↗

In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. In 1989, the Loma Prieta earthquake also affected the San Francisco area, and measured 6.9 on the Richter scale. Compare the intensities of the two earthquakes.

Answer

The intensity of the 1906 earthquake was about 8 times the intensity of the 1989 earthquake.

Try It 10.54 §10.3 source ↗

In 2014, Chile experienced an intense earthquake with a magnitude of 8.2 on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured 5.1 on the Richter scale. Compare the intensities of the two earthquakes.

Answer

The intensity of the earthquake in Chile was about 1,259 times the intensity of the earthquake in Los Angeles.

10.3-127 §10.3 source ↗

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

$2^{5} = 32$

Answer

{log}_{2} 32 = 5

10.3-129 §10.3 source ↗

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

$5^{3} = 125$

Answer

{log}_{5} 125 = 3

10.3-131 §10.3 source ↗

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

$10^{−2} = \frac{1}{100}$

Answer

log \frac{1}{100} = −2

10.3-133 §10.3 source ↗

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

$x^{\frac{1}{3}} = \sqrt[3]{6}$

Answer

{log}_{x} \sqrt[3]{6} = \frac{1}{3}

10.3-135 §10.3 source ↗

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

$17^{x} = \sqrt[5]{17}$

Answer

{log}_{17} \sqrt[5]{17} = x

10.3-137 §10.3 source ↗

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

${(\frac{1}{3})}^{4} = \frac{1}{81}$

Answer

{log}_{\frac{1}{3}} \frac{1}{81} = 4

10.3-139 §10.3 source ↗

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

$4^{−3} = \frac{1}{64}$

Answer

{log}_{4} \frac{1}{64} = −3

10.3-141 §10.3 source ↗

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

$e^{3} = x$

Answer

ln x = 3

10.3-143 §10.3 source ↗

In the following exercises, convert each logarithmic equation to exponential form.

$6 = {log}_{2} 64$

Answer

64 = 2^{6}

10.3-145 §10.3 source ↗

In the following exercises, convert each logarithmic equation to exponential form.

$5 = {log}_{x} 32$

Answer

32 = x^{5}

10.3-147 §10.3 source ↗

In the following exercises, convert each logarithmic equation to exponential form.

$0 = {log}_{7} 1$

Answer

1 = 7^{0}

10.3-149 §10.3 source ↗

In the following exercises, convert each logarithmic equation to exponential form.

$1 = {log}_{9} 9$

Answer

9 = 9^{1}

10.3-151 §10.3 source ↗

In the following exercises, convert each logarithmic equation to exponential form.

$3 =$ ${log}_{10} 1,000$

Answer

1,000 = 10^{3}

10.3-153 §10.3 source ↗

In the following exercises, convert each logarithmic equation to exponential form.

$x = {log}_{e} 43$

Answer

43 = e^{x}

10.3-155 §10.3 source ↗

Evaluate Logarithmic Functions

In the following exercises, find the value of $x$ in each logarithmic equation.

${log}_{x} 121 = 2$

Answer

x = 11

10.3-157 §10.3 source ↗

Evaluate Logarithmic Functions

In the following exercises, find the value of $x$ in each logarithmic equation.

${log}_{x} 64 = 3$

Answer

x = 4

10.3-159 §10.3 source ↗

Evaluate Logarithmic Functions

In the following exercises, find the value of $x$ in each logarithmic equation.

${log}_{5} x = 3$

Answer

x = 125

10.3-161 §10.3 source ↗

Evaluate Logarithmic Functions

In the following exercises, find the value of $x$ in each logarithmic equation.

${log}_{3} x = −5$

Answer

x = \frac{1}{243}

10.3-163 §10.3 source ↗

Evaluate Logarithmic Functions

In the following exercises, find the value of $x$ in each logarithmic equation.

${log}_{\frac{1}{3}} \frac{1}{9} = x$

Answer

x = 2

10.3-165 §10.3 source ↗

Evaluate Logarithmic Functions

In the following exercises, find the value of $x$ in each logarithmic equation.

${log}_{\frac{1}{9}} 81 = x$

Answer

x = −2

10.3-167 §10.3 source ↗

In the following exercises, find the exact value of each logarithm without using a calculator.

${log}_{6} 36$

Answer

10.3-169 §10.3 source ↗

In the following exercises, find the exact value of each logarithm without using a calculator.

${log}_{5} 1$

Answer

10.3-171 §10.3 source ↗

In the following exercises, find the exact value of each logarithm without using a calculator.

${log}_{27} 3$

Answer

\frac{1}{3}

10.3-173 §10.3 source ↗

In the following exercises, find the exact value of each logarithm without using a calculator.

${log}_{\frac{1}{2}} 4$

Answer

−2

10.3-175 §10.3 source ↗

In the following exercises, find the exact value of each logarithm without using a calculator.

${log}_{3} \frac{1}{27}$

Answer

−3

10.3-177 §10.3 source ↗

In the following exercises, find the exact value of each logarithm without using a calculator.

${log}_{9} \frac{1}{81}$

Answer

−2

10.3-179 §10.3 source ↗

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

$y = {log}_{4} x$

Answer

This figure shows the logarithmic curve going through the points (1 over 4, negative 1), (1, 0), and (4, 1).

10.3-181 §10.3 source ↗

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

$y = {log}_{7} x$

Answer

This figure shows that the logarithmic curve going through the points (1 over 7, negative 1), (1, 0), and (7, 1).

10.3-183 §10.3 source ↗

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

$y = {log}_{2.5} x$

Answer

This figure shows the logarithmic curve going through the points (2 over 5, negative 1), (1, 0), and (2.5, 1).

10.3-185 §10.3 source ↗

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

$y = {log}_{\frac{1}{5}} x$

Answer

This figure shows the logarithmic curve going through the points (1 over 5, 1), (1, 0), and (5, negative 1).

10.3-187 §10.3 source ↗

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

$y = {log}_{0.6} x$

Answer

This figure shows the logarithmic curve going through the points (3 over 5, 1), (1, 0), and (5 over 3, negative 1).

10.3-189 §10.3 source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

${log}_{a} 81 = 2$

Answer

a = 9

10.3-191 §10.3 source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

${log}_{a} 27 = 3$

Answer

a = 3

10.3-193 §10.3 source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

${log}_{a} 24 = 3$

Answer

a = 2 \sqrt[3]{3}

10.3-195 §10.3 source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

$ln x = 4$

Answer

x = e^{4}

10.3-197 §10.3 source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

${log}_{2} (6 x + 2) =$ $5$

Answer

x = 5

10.3-199 §10.3 source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

${log}_{3} (5 x - 4) =$ $4$

Answer

x = 17

10.3-201 §10.3 source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

${log}_{4} (3 x - 2) =$ $2$

Answer

x = 6

10.3-203 §10.3 source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

$ln e^{2 x} = 6$

Answer

x = 3

10.3-205 §10.3 source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

$log (x^{2} - 25) =$ $2$

Answer

x = −5 \sqrt{5}, x = 5 \sqrt{5}

10.3-207 §10.3 source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

${log}_{3} (x^{2} + 2) =$ $3$

Answer

x = −5, x = 5

10.3-209 §10.3 source ↗

Use Logarithmic Models in Applications

In the following exercises, use a logarithmic model to solve.

What is the decibel level of a whisper with intensity $10^{−10}$ watts per square inch?

Answer

A whisper has a decibel level of 20 dB.

10.3-211 §10.3 source ↗

Use Logarithmic Models in Applications

In the following exercises, use a logarithmic model to solve.

What is the decibel level of the sound of a garbage disposal with intensity $10^{−2}$ watts per square inch?

Answer

The sound of a garbage disposal has a decibel level of 100 dB.

10.3-213 §10.3 source ↗

Use Logarithmic Models in Applications

In the following exercises, use a logarithmic model to solve.

The Los Angeles area experiences many earthquakes. In 1994, the Northridge earthquake measured magnitude of $6.7$ on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured $5.1$ on the Richter scale. Compare the intensities of the two earthquakes.

Answer

The intensity of the 1994 Northridge earthquake in the Los Angeles area was about 40 times the intensity of the 2014 earthquake.

10.3-215 §10.3 source ↗

Use Logarithmic Models in Applications

In the following exercises, use a logarithmic model to solve.

Explain the difference between common logarithms and natural logarithms.

Answer

Answers will vary.

10.3-217 §10.3 source ↗

Use Logarithmic Models in Applications

In the following exercises, use a logarithmic model to solve.

Explain how to find the ${log}_{7} 32$ on your calculator.

Answer

Answers will vary.

Be Prepared 10.10 §10.4 source ↗

Evaluate: ⓐ

a^{0}

ⓑ

a^{1} .

If you missed this problem, review Example 5.14.

Example 5.14 in the book ↗

Answer

ⓐ

1

; ⓑ

a

Be Prepared 10.11 §10.4 source ↗

Write with a rational exponent:

\sqrt[3]{x^{2} y} .

If you missed this problem, review Example 8.27.

Example 8.27 in the book ↗

Answer

{x^{2} y}^{\frac{1}{3}}

Be Prepared 10.12 §10.4 source ↗

Round to three decimal places: 2.5646415.
If you missed this problem, review Example 1.34.

Example 1.34 in the book ↗

Answer

2.565

Try It 10.55 §10.4 source ↗

Evaluate using the properties of logarithms: ⓐ

{log}_{13} 1

ⓑ

{log}_{9} 9 .

Answer

ⓐ 0 ⓑ 1

Try It 10.56 §10.4 source ↗

Evaluate using the properties of logarithms: ⓐ

{log}_{5} 1

ⓑ

{log}_{7} 7 .

Answer

ⓐ 0 ⓑ 1

Try It 10.57 §10.4 source ↗

Evaluate using the properties of logarithms: ⓐ

5^{{log}_{5} 15}

ⓑ

{log}_{7} 7^{4} .

Answer

ⓐ 15 ⓑ 4

Try It 10.58 §10.4 source ↗

Evaluate using the properties of logarithms: ⓐ

2^{{log}_{2} 8}

ⓑ

{log}_{2} 2^{15} .

Answer

ⓐ 8 ⓑ 15

Try It 10.59 §10.4 source ↗

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

ⓐ ${log}_{3} 3 x$ ⓑ ${log}_{2} 8 x y$

Answer

ⓐ

1 + {log}_{3} x

ⓑ

3 +

{log}_{2} x +

{log}_{2} y

Try It 10.60 §10.4 source ↗

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

ⓐ ${log}_{9} 9 x$ ⓑ ${log}_{3} 27 x y$

Answer

ⓐ

1 + {log}_{9} x

ⓑ

3 +

{log}_{3} x +

{log}_{3} y

Try It 10.61 §10.4 source ↗

Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.

ⓐ ${log}_{4} \frac{3}{4}$ ⓑ $log \frac{x}{1000}$

Answer

ⓐ

{log}_{4} 3 - 1

ⓑ

log x - 3

Try It 10.62 §10.4 source ↗

Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.

ⓐ ${log}_{2} \frac{5}{4}$ ⓑ $log \frac{10}{y}$

Answer

ⓐ

{log}_{2} 5 - 2

ⓑ

1 - log y

Try It 10.63 §10.4 source ↗

Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.

ⓐ ${log}_{7} 5^{4}$ ⓑ $log x^{100}$

Answer

ⓐ

4 {log}_{7} 5

ⓑ

100 \cdot log x

Try It 10.64 §10.4 source ↗

Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.

ⓐ ${log}_{2} 3^{7}$ ⓑ $log x^{20}$

Answer

ⓐ

7 {log}_{2} 3

ⓑ

20 \cdot log x

Try It 10.65 §10.4 source ↗

Use the Properties of Logarithms to expand the logarithm

{log}_{2} (5 x^{4} y^{2})

. Simplify, if possible.

Answer

{log}_{2} 5 +

4 {log}_{2} x +

2 {log}_{2} y

Try It 10.66 §10.4 source ↗

Use the Properties of Logarithms to expand the logarithm

{log}_{3} (7 x^{5} y^{3})

. Simplify, if possible.

Answer

{log}_{3} 7 +

5 {log}_{3} x +

3 {log}_{3} y

Try It 10.67 §10.4 source ↗

Use the Properties of Logarithms to expand the logarithm

{log}_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}}

. Simplify, if possible.

Answer

\frac{1}{5} (4 {log}_{4} x - \frac{1}{2} - 3 {log}_{4} y - 2 {log}_{4} z)

Try It 10.68 §10.4 source ↗

Use the Properties of Logarithms to expand the logarithm

{log}_{3} \sqrt[3]{\frac{x^{2}}{5 y^{} z}}

. Simplify, if possible.

Answer

\frac{1}{3} (2 {log}_{3} x - {log}_{3} 5 - {log}_{3} y - {log}_{3} z)

Try It 10.69 §10.4 source ↗

Use the Properties of Logarithms to condense the logarithm

{log}_{2} 5 +

{log}_{2} x -

{log}_{2} y

. Simplify, if possible.

Answer

{log}_{2} \frac{5 x}{y}

Try It 10.70 §10.4 source ↗

Use the Properties of Logarithms to condense the logarithm

{log}_{3} 6 -

{log}_{3} x -

{log}_{3} y

. Simplify, if possible.

Answer

{log}_{3} \frac{6}{x y}

Try It 10.71 §10.4 source ↗

Use the Properties of Logarithms to condense the logarithm

3 {log}_{2} x +

2 {log}_{2} (x - 1)

. Simplify, if possible.

Answer

{log}_{2} x^{3} {(x - 1)}^{2}

Try It 10.72 §10.4 source ↗

Use the Properties of Logarithms to condense the logarithm

2 log x +

2 log (x + 1)

. Simplify, if possible.

Answer

log x^{2} {(x + 1)}^{2}

Try It 10.73 §10.4 source ↗

Rounding to three decimal places, approximate

{log}_{3} 42 .

Answer

3.402

Try It 10.74 §10.4 source ↗

Rounding to three decimal places, approximate

{log}_{5} 46 .

Answer

2.379

10.4-219 §10.4 source ↗

Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

ⓐ ${log}_{12} 1$ ⓑ $ln e$

Answer

ⓐ 0 ⓑ 1

10.4-221 §10.4 source ↗

Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

ⓐ $5^{{log}_{5} 10}$ ⓑ ${log}_{4} 4^{10}$

Answer

ⓐ 10 ⓑ 10

10.4-223 §10.4 source ↗

Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

ⓐ $6^{{log}_{6} 15}$ ⓑ ${log}_{8} 8^{−4}$

Answer

ⓐ 15 ⓑ

−4

10.4-225 §10.4 source ↗

Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

ⓐ $10^{log \sqrt{3}}$ ⓑ $log 10^{−1}$

Answer

ⓐ

\sqrt{3}

ⓑ

−1

10.4-227 §10.4 source ↗

Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

ⓐ $e^{ln 3}$ ⓑ $ln e^{7}$

Answer

ⓐ 3 ⓑ 7

10.4-229 §10.4 source ↗

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

${log}_{5} 8 y$

Answer

{log}_{5} 8 + {log}_{5} y

10.4-231 §10.4 source ↗

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

${log}_{3} 81 x y$

Answer

4 +

{log}_{3} x +

{log}_{3} y

10.4-233 §10.4 source ↗

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$log 1000 y$

Answer

3 + log y

10.4-235 §10.4 source ↗

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

${log}_{6} \frac{5}{6}$

Answer

{log}_{6} 5 - 1

10.4-237 §10.4 source ↗

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

${log}_{5} \frac{125}{x}$

Answer

3 - {log}_{5} x

10.4-239 §10.4 source ↗

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$log \frac{10,000}{y}$

Answer

4 - log y

10.4-241 §10.4 source ↗

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$ln \frac{e^{4}}{16}$

Answer

4 - ln 16

10.4-243 §10.4 source ↗

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.

${log}_{2} x^{5}$

Answer

5 {log}_{2} x

10.4-245 §10.4 source ↗

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.

$log x^{−3}$

Answer

−3 log x

10.4-247 §10.4 source ↗

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.

${log}_{5} \sqrt[3]{x}$

Answer

\frac{1}{3} {log}_{5} x

10.4-249 §10.4 source ↗

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.

$ln x^{\sqrt[3]{4}}$

Answer

\sqrt[3]{4} ln x

10.4-251 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

${log}_{2} (3 x^{5} y^{3})$

Answer

{log}_{2} 3 +

5 {log}_{2} x +

3 {log}_{2} y

10.4-253 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

${log}_{5} (\sqrt[4]{21} y^{3})$

Answer

\frac{1}{4} {log}_{5} 21 +

3 {log}_{5} y

10.4-255 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

${log}_{5} \frac{4 a b^{3} c^{4}}{d^{2}}$

Answer

{log}_{5} 4 +

{log}_{5} a +

3 {log}_{5} b

+

4 {log}_{5} c -

2 {log}_{5} d

10.4-257 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

${log}_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}$

Answer

\frac{2}{3} {log}_{3} x -

3 -

4 {log}_{3} y

10.4-259 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

${log}_{3} \frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}}$

Answer

\frac{1}{2} {log}_{3} (3 x + 2 y^{2}) -

{log}_{3} 5 -

2 {log}_{3} z

10.4-261 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

${log}_{5} \sqrt[3]{\frac{3 x^{2}}{4 y^{3} z}}$

Answer

\frac{1}{3} ({log}_{5} 3 + 2 {log}_{5} x - {log}_{5} 4

-

3 {log}_{5} y -

{log}_{5} z)

10.4-263 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$log 4 + log 25$

Answer

10.4-265 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

${log}_{3} 36 -$ ${log}_{3} 4$

Answer

10.4-267 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

${log}_{2} 5 -$ ${log}_{2} (x - 1)$

Answer

{log}_{2} \frac{5}{x - 1}

10.4-269 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

${log}_{5} 2 -$ ${log}_{5} x -$ ${log}_{5} y$

Answer

{log}_{5} \frac{2}{x y}

10.4-271 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$6 {log}_{3} x +$ $9 {log}_{3} y$

Answer

{log}_{3} x^{6} y^{9}

10.4-273 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$log (x^{2} + 2 x + 1) -$ $2 log (x + 1)$

Answer

10.4-275 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$3 ln x +$ $4 ln y -$ $2 ln z$

Answer

ln \frac{x^{3} y^{4}}{z^{2}}

10.4-277 §10.4 source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$2 log (2 x + 3) +$ $\frac{1}{2} log (x + 1)$

Answer

log {(2 x + 3)}^{2} \cdot \sqrt{x + 1}

10.4-279 §10.4 source ↗

Use the Change-of-Base Formula

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

${log}_{5} 46$

Answer

2.379

10.4-281 §10.4 source ↗

Use the Change-of-Base Formula

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

${log}_{15} 93$

Answer

1.674

10.4-283 §10.4 source ↗

Use the Change-of-Base Formula

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

${log}_{\sqrt{3}} 21$

Answer

5.542

10.4-285 §10.4 source ↗

Use the Change-of-Base Formula

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

Write the Power Property in your own words. Does it apply to each of the following? ${log}_{a} x^{p},$ ${({log}_{a} x)}^{r} .$ Why or why not?

Answer

Answers will vary.

10.4-287 §10.4 source ↗

Use the Change-of-Base Formula

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

Explain how to find the value of ${log}_{7} 15$ using your calculator.

Answer

Answers will vary.

Be Prepared 10.13 §10.5 source ↗

Solve:

x^{2} = 16 .

If you missed this problem, review Example 6.46.

Example 6.46 in the book ↗

Answer

x = 4, x = - 4

Be Prepared 10.14 §10.5 source ↗

Solve:

x^{2} - 5 x + 6 = 0 .

If you missed this problem, review Example 6.45.

Example 6.45 in the book ↗

Answer

x = 2, x = 3

Be Prepared 10.15 §10.5 source ↗

Solve:

x (x + 6) =

2 x +

5 .

If you missed this problem, review Example 6.47.

Example 6.47 in the book ↗

Answer

x = - 5, x = 1

Try It 10.75 §10.5 source ↗

Solve:

2 {log}_{3} x =

{log}_{3} 36

Answer

x = 6

Try It 10.76 §10.5 source ↗

Solve:

3 log x = log 64

Answer

x = 4

Try It 10.77 §10.5 source ↗

Solve:

{log}_{2} x +

{log}_{2} (x - 2) =

3

Answer

x = 4

Try It 10.78 §10.5 source ↗

Solve:

{log}_{2} x +

{log}_{2} (x - 6) =

4

Answer

x = 8

Try It 10.79 §10.5 source ↗

Solve:

log (x + 2) -

log (4 x + 3) =

− log x .

Answer

x = 3

Try It 10.80 §10.5 source ↗

Solve:

log (x - 2) -

log (4 x + 16) =

log \frac{1}{x} .

Answer

x = 8

Try It 10.81 §10.5 source ↗

Solve

7^{x} = 43 .

Find the exact answer and then approximate it to three decimal places.

Answer

x =

\frac{log 43}{log 7} \approx

1.933

Try It 10.82 §10.5 source ↗

Solve

8^{x} = 98 .

Find the exact answer and then approximate it to three decimal places.

Answer

x =

\frac{log 98}{log 8} \approx

2.205

Try It 10.83 §10.5 source ↗

Solve

2 e^{x - 2} = 18 .

Find the exact answer and then approximate it to three decimal places.

Answer

x =

ln 9 +

2 \approx

4.197

Try It 10.84 §10.5 source ↗

Solve

5 e^{2 x} = 25 .

Find the exact answer and then approximate it to three decimal places.

Answer

x =

\frac{ln 5}{2} \approx

0.805

Try It 10.85 §10.5 source ↗

Hector invests

$ 10,000

at age 21. He hopes the investments will be worth

$ 150,000

when he turns 50. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal?

Answer

r \approx 9.3 %

Try It 10.86 §10.5 source ↗

Rachel invests

$ 15,000

at age 25. She hopes the investments will be worth

$ 90,000

when she turns 40. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?

Answer

r \approx 11.9 %

Try It 10.87 §10.5 source ↗

Researchers recorded that a certain bacteria population grew from 100 to 500 in 6 hours. At this rate of growth, how many bacteria will there be 24 hours from the start of the experiment?

Answer

There will be 62,500 bacteria.

Try It 10.88 §10.5 source ↗

Researchers recorded that a certain bacteria population declined from 700,000 to 400,000 in 5 hours after the administration of medication. At this rate of decay, how many bacteria will there be 24 hours from the start of the experiment?

Answer

There will be 47,700 bacteria.

Try It 10.89 §10.5 source ↗

The half-life of magnesium-27 is 9.45 minutes. How much of a 10-mg sample will be left in 6 minutes?

Answer

There will be 6.44 mg left.

Try It 10.90 §10.5 source ↗

The half-life of radioactive iodine is 60 days. How much of a 50-mg sample will be left in 40 days?

Answer

There will be 31.5 mg left.

10.5-289 §10.5 source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

$log 49 = 2 log x$

Answer

x = 7

10.5-291 §10.5 source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

$3 {log}_{6} x =$ ${log}_{6} 64$

Answer

x = 4

10.5-293 §10.5 source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

${log}_{3} (x^{2} + 3) =$ ${log}_{3} 4 x$

Answer

x = 1,

x = 3

10.5-295 §10.5 source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

${log}_{4} x +$ ${log}_{4} x =$ $3$

Answer

x = 8

10.5-297 §10.5 source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

${log}_{3} x +$ ${log}_{3} (x + 6) =$ $3$

Answer

x = 3

10.5-299 §10.5 source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

$log x +$ $log (x - 15) =$ $2$

Answer

x = 20

10.5-301 §10.5 source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

$log (x - 1) -$ $log (x + 3) =$ $log \frac{1}{x}$

Answer

x = 3

10.5-303 §10.5 source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

${log}_{5} (x + 1) +$ ${log}_{5} (x - 5) =$ ${log}_{5} 7$

Answer

x = 6

10.5-305 §10.5 source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

$log (5 x + 1) =$ $log (x + 3) +$ $log 2$

Answer

x = \frac{5}{3}

10.5-307 §10.5 source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$2^{x} = 74$

Answer

x =

\frac{log 74}{log 2} \approx

6.209

10.5-309 §10.5 source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$4^{x} = 112$

Answer

x =

\frac{log 112}{log 4} \approx

3.404

10.5-311 §10.5 source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$e^{x} = 8$

Answer

x = ln 8 \approx 2.079

10.5-313 §10.5 source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

${(\frac{1}{3})}^{x} = 8$

Answer

x =

\frac{log 8}{log \frac{1}{3}} \approx

− 1.893

10.5-315 §10.5 source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$3 e^{x + 2} = 9$

Answer

x =

ln 3 -

2 \approx

− 0.901

10.5-317 §10.5 source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$2 e^{3 x} = 32$

Answer

x =

\frac{ln 16}{3} \approx

0.924

10.5-319 §10.5 source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$\frac{1}{3} e^{x} = 2$

Answer

x = ln 6 \approx 1.792

10.5-321 §10.5 source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$e^{x - 1} + 4 = 12$

Answer

x =

ln 8 +

1 \approx

3.079

10.5-323 §10.5 source ↗

In the following exercises, solve each equation.

$6^{4 x - 17} = 216$

Answer

x = 5

10.5-325 §10.5 source ↗

In the following exercises, solve each equation.

$\frac{e^{x^{2}}}{e^{x}} = e^{20}$

Answer

x = −4, x = 5

10.5-327 §10.5 source ↗

In the following exercises, solve each equation.

${log}_{a} 81 = 4$

Answer

a = 3

10.5-329 §10.5 source ↗

In the following exercises, solve each equation.

$ln x = 9$

Answer

x = e^{9}

10.5-331 §10.5 source ↗

In the following exercises, solve each equation.

${log}_{4} (7 x + 15) =$ $3$

Answer

x = 7

10.5-333 §10.5 source ↗

In the following exercises, solve each equation.

$ln e^{6 x} = 18$

Answer

x = 3

10.5-335 §10.5 source ↗

In the following exercises, solve each equation.

$7 {log}_{3} x =$ ${log}_{3} 128$

Answer

x = 2

10.5-337 §10.5 source ↗

In the following exercises, solve each equation.

${log}_{9} x +$ ${log}_{9} (x - 4) =$ ${log}_{9} 12$

Answer

x = 6

10.5-339 §10.5 source ↗

In the following exercises, solve each equation.

${log}_{6} (x + 1) -$ ${log}_{6} (4 x + 10) =$ ${log}_{6} \frac{1}{x}$

Answer

x = 5

10.5-341 §10.5 source ↗

In the following exercises, solve for x, giving an exact answer as well as an approximation to three decimal places.

${(\frac{1}{2})}^{x} = 10$

Answer

x =

\frac{log 10}{log \frac{1}{2}} \approx

− 3.322

10.5-343 §10.5 source ↗

In the following exercises, solve for x, giving an exact answer as well as an approximation to three decimal places.

$8 e^{x + 5} = 56$

Answer

x =

ln 7 -

5 \approx

− 3.054

10.5-345 §10.5 source ↗

Use Exponential Models in Applications

In the following exercises, solve.

Alice invests $$ 15,000$ at age 30 from the signing bonus of her new job. She hopes the investments will be worth $$ 30,000$ when she turns 40. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?

Answer

6.9 %

10.5-347 §10.5 source ↗

Use Exponential Models in Applications

In the following exercises, solve.

Simone invests $$ 8,000$ in an account that compounds interest quarterly and earns $5 % .$ How long will it take for his money to double?

Answer

13.9 years

10.5-349 §10.5 source ↗

Use Exponential Models in Applications

In the following exercises, solve.

Researchers recorded that a certain bacteria population declined from 800,000 to 500,000 in 6 hours after the administration of medication. At this rate of decay, how many bacteria will there be in 24 hours?

Answer

122,070 bacteria

10.5-351 §10.5 source ↗

Use Exponential Models in Applications

In the following exercises, solve.

A bacteria doubles its original population in 24 hours $(A = 2 A_{0}) .$ How big will its population be in 72 hours?

Answer

8 times as large as the original population

10.5-353 §10.5 source ↗

Use Exponential Models in Applications

In the following exercises, solve.

Radioactive technetium-99m is often used in diagnostic medicine as it has a relatively short half-life but lasts long enough to get the needed testing done on the patient. If its half-life is 6 hours, how much of the radioactive material form a 0.5 ml injection will be in the body in 24 hours?

Answer

0.03 ml

10.5-355 §10.5 source ↗

Use Exponential Models in Applications

In the following exercises, solve.

What is the difference between the equation for exponential growth versus the equation for exponential decay?

Answer

Answers will vary.

10.R-357 §10.R source ↗

Find and Evaluate Composite Functions

$f (x) = 4 x$ and
$g (x) = x^{2} + 3 x$

Answer

ⓐ

4 x^{2} + 12 x

ⓑ

16 x^{2} + 12 x

ⓒ

4 x^{3} + 12 x^{2}

10.R-359 §10.R source ↗

In the following exercises, evaluate the composition.

For functions
$f (x) = 2 x^{3} + 5$ and
$g (x) = 3 x^{2} - 7,$ find
ⓐ $(f \circ g) (−1)$
ⓑ $(g \circ f) (−2)$
ⓒ $(g \circ g) (1)$

Answer

ⓐ

−123

ⓑ 356 ⓒ 41

10.R-361 §10.R source ↗

Determine Whether a Function is One-to-One

In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

${(−3, 0), (−2, −2), (−1, 0), (0, 1),$
$(1, 2),$ $(2, 1),$ $(3, −1)}$

Answer

Function; not one-to-one

10.R-363 §10.R source ↗

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

ⓐ

This figure shows a line from (negative 6, negative 2) up to (negative 1, 3) and then down from there to (6, negative 4).

ⓑ

This figure shows a line from (6, 5) down to (0, negative 1) and then down from there to (5, negative 6).

Answer

ⓐ Function; not one-to-one ⓑ Not a function

10.R-365 §10.R source ↗

Find the Inverse of a Function

In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.

${(−3, 10), (−2, 5), (−1, 2), (0, 1)}$

Answer

Inverse function:

{(10, −3), (5, −2), (2, −1), (1, 0)} .

Domain:

{1, 2, 5, 10} .

Range:

{−3, −2, −1, 0} .

10.R-367 §10.R source ↗

In the following exercises, verify that the functions are inverse functions.

$f (x) = 3 x + 7$ and
$g (x) = \frac{x - 7}{3}$

Answer

g (f (x)) = x,

and

f (g (x)) = x,

so they are inverses.

10.R-369 §10.R source ↗

In the following exercises, find the inverse of each function.

$f (x) = 6 x - 11$

Answer

f^{−1} (x) =

\frac{x + 11}{6}

10.R-371 §10.R source ↗

In the following exercises, find the inverse of each function.

$f (x) = \frac{1}{x + 5}$

Answer

f^{−1} (x) = \frac{1}{x} - 5

10.R-373 §10.R source ↗

Graph Exponential Functions

In the following exercises, graph each of the following functions.

$f (x) = 4^{x}$

Answer

This figure shows an exponential line passing through the points (negative 1, 1 over 4), (0, 1), and (1, 4).

10.R-375 §10.R source ↗

Graph Exponential Functions

In the following exercises, graph each of the following functions.

$g (x) =$ ${(0.75)}^{x}$

Answer

This figure shows an exponential line passing through the points (negative 1, 4 over 3), (0, 1), and (1, 3 over 4).

10.R-377 §10.R source ↗

Graph Exponential Functions

In the following exercises, graph each of the following functions.

$f (x) =$ ${(2.3)}^{x} -$ $3$

Answer

This figure shows an exponential line passing through the points (negative 1, negative 59 over 23), (0, negative 2), and (1, negative7 over 10).

10.R-379 §10.R source ↗

Graph Exponential Functions

In the following exercises, graph each of the following functions.

$f (x) = − e^{x}$

Answer

This figure shows an exponential line passing through the points (negative 1, negative 1 over e), (0, negative 1), and (1, negative e).

10.R-381 §10.R source ↗

Solve Exponential Equations

In the following exercises, solve each equation.

$2^{x^{2}} = 16$

Answer

x = −2, x = 2

10.R-383 §10.R source ↗

Solve Exponential Equations

In the following exercises, solve each equation.

$5^{x^{2} + 2 x} = \frac{1}{5}$

Answer

x = −1

10.R-385 §10.R source ↗

Solve Exponential Equations

In the following exercises, solve each equation.

$\frac{e^{x^{2}}}{e^{15}} = e^{2 x}$

Answer

x = −3, x = 5

10.R-387 §10.R source ↗

Use Exponential Models in Applications

In the following exercises, solve.

Sayed deposits $$ 20,000$ in an investment account. What will be the value of his investment in 30 years if the investment is earning $7 %$ per year and is compounded continuously?

Answer

$ 163,323.40

10.R-389 §10.R source ↗

Use Exponential Models in Applications

In the following exercises, solve.

In the last five years the population of the United States has grown at a rate of $0.7 %$ per year to about 318,900,000. If this rate continues, what will be the population in 5 more years?

Answer

330,259,000

10.R-391 §10.R source ↗

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

$10^{−3} = \frac{1}{1,000}$

Answer

log \frac{1}{1,000} =

−3

10.R-393 §10.R source ↗

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

$e^{y} = 16$

Answer

ln 16 = y

10.R-395 §10.R source ↗

In the following exercises, convert each logarithmic equation to exponential form.

$5 =$ $log 100,000$

Answer

100000 = 10^{5}

10.R-397 §10.R source ↗

Evaluate Logarithmic Functions

In the following exercises, solve for x.

${log}_{x} 125 = 3$

Answer

x = 5

10.R-399 §10.R source ↗

Evaluate Logarithmic Functions

In the following exercises, solve for x.

${log}_{\frac{1}{2}} \frac{1}{16} = x$

Answer

x = 4

10.R-401 §10.R source ↗

In the following exercises, find the exact value of each logarithm without using a calculator.

${log}_{8} 1$

Answer

10.R-403 §10.R source ↗

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

$y = {log}_{5} x$

Answer

This figure shows a logarithmic line passing through the points (1 over 5, negative 1), (1, 0), and (5, 1).

10.R-405 §10.R source ↗

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

$y = {log}_{0.8} x$

Answer

This figure shows a logarithmic line passing through the points (4 over 5, 1), (1, 0), and (5 over 4, negative 1).

10.R-407 §10.R source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

$ln x = −3$

Answer

x = e^{−3}

10.R-409 §10.R source ↗

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

$ln e^{3 x} = 24$

Answer

x = 8

10.R-411 §10.R source ↗

Use Logarithmic Models in Applications

What is the decibel level of a train whistle with intensity $10^{−3}$ watts per square inch?

Answer

90 dB

10.R-413 §10.R source ↗

Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

ⓐ $5^{{log}_{5} 13}$ ⓑ ${log}_{3} 3^{−9}$

Answer

ⓐ 13 ⓑ

−9

10.R-415 §10.R source ↗

Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

ⓐ $e^{ln 8}$ ⓑ $ln e^{5}$

Answer

ⓐ 8 ⓑ 5

10.R-417 §10.R source ↗

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$log 10,000 m$

Answer

4 + log m

10.R-419 §10.R source ↗

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

$ln \frac{e^{5}}{2}$

Answer

5 - ln 2

10.R-421 §10.R source ↗

In the following exercises, use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.

${log}_{4} \sqrt[7]{z}$

Answer

\frac{1}{7} {log}_{4} z

10.R-423 §10.R source ↗

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\log_{5} \frac{8 a^{2} b^{6} c}{d^{3}}$

Answer

{log}_{5} 8 +

2 {log}_{5} a +

6 {log}_{5} b

+

{log}_{5} c -

3 {log}_{5} d

10.R-425 §10.R source ↗

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

${log}_{6} \sqrt[3]{\frac{7 x^{2}}{6 y^{3} z^{5}}}$

Answer

\frac{1}{3} ({log}_{6} 7 + 2 {log}_{6} x - 1 - 3 {log}_{6} y

- 5 {log}_{6} z)

10.R-427 §10.R source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$3 {log}_{3} x +$ $7 {log}_{3} y$

Answer

{log}_{3} x^{3} y^{7}

10.R-429 §10.R source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$\frac{1}{4} log y -$ $2 log (y - 3)$

Answer

log \frac{\sqrt[4]{y}}{{(y - 3)}^{2}}

10.R-431 §10.R source ↗

Use the Change-of-Base Formula

In the following exercises, rounding to three decimal places, approximate each logarithm.

${log}_{\sqrt{3}} 16$

Answer

5.047

10.R-433 §10.R source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

${log}_{2} x +$ ${log}_{2} (x - 2) =$ $3$

Answer

x = 4

10.R-435 §10.R source ↗

Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for x.

${log}_{4} (x - 2) +$ ${log}_{4} (x + 5) =$ ${log}_{4} 8$

Answer

x = 3

10.R-437 §10.R source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$2^{x} = 101$

Answer

x =

\frac{log 101}{log 2} \approx

6.658

10.R-439 §10.R source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

${(\frac{1}{3})}^{x} = 7$

Answer

x =

\frac{log 7}{log \frac{1}{3}} \approx

− 1.771

10.R-441 §10.R source ↗

Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$e^{x - 4} + 8 = 23$

Answer

x =

ln 15 +

4 \approx

6.708

10.R-443 §10.R source ↗

Use Exponential Models in Applications

Elise invests $$ 4500$ in an account that compounds interest monthly and earns $6 % .$ How long will it take for her money to double?

Answer

11.6 years

10.R-445 §10.R source ↗

Use Exponential Models in Applications

Mouse populations can double in 8 months $(A = 2 A_{0}) .$ How long will it take for a mouse population to triple?

Answer

12.7 months

10.T-447 §10.T source ↗

For the functions,

f (x) = 6 x + 1

and

g (x) = 8 x - 3,

find ⓐ

(f \circ g) (x),

ⓑ

(g \circ f) (x),

and ⓒ

(f \cdot g) (x) .

Answer

ⓐ

48 x - 17

ⓑ

48 x + 5

ⓒ

48 x^{2} - 10 x - 3

10.T-449 §10.T source ↗

Determine whether each graph is the graph of a function and if so, is it one-to-one.

ⓐ

This figure shows a parabola opening to the right with vertex (negative 3, 0).

ⓑ

This figure shows an exponential line passing through the points (negative 1, 1 over 2), (0, 1), and (1, 2).

Answer

ⓐ Not a function ⓑ One-to-one function

10.T-451 §10.T source ↗

Find the inverse of the function

f (x) = x^{5} - 9 .

Answer

f^{−1} (x) = \sqrt[5]{x + 9}

10.T-453 §10.T source ↗

Solve the equation

2^{2 x - 4} = 64 .

Answer

x = 5

10.T-455 §10.T source ↗

Megan invested

$ 21,000

in a savings account. If the interest rate is

5 %,

how much will be in the account in 8 years by each method of compounding?
ⓐ compound quarterly
ⓑ compound monthly
ⓒ compound continuously.

Answer

ⓐ

$ 31,250.74

ⓑ

$ 31,302.29

ⓒ

$ 31,328.32

10.T-457 §10.T source ↗

Convert the equation from logarithmic equation to exponential form:

3 = {log}_{7} 343

Answer

343 = 7^{3}

10.T-459 §10.T source ↗

Evaluate

{log}_{11} 1 .

Answer

10.T-461 §10.T source ↗

Graph the function

y = {log}_{3} x .

Answer

This figure shows a logarithmic line passing through (1 over 3, 1), (1, 0), and (3, 1).

10.T-463 §10.T source ↗

What is the decibel level of a small fan with intensity

10^{−8}

watts per square inch?

Answer

40 dB

10.T-465 §10.T source ↗

In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.

${log}_{5} 25 a b$

Answer

2 +

{log}_{5} a +

{log}_{5} b

10.T-467 §10.T source ↗

In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.

${log}_{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}}$

Answer

\frac{1}{4} ({log}_{2} 5 + 3 {log}_{2} x - 4 - 2 {log}_{2} y

- 7 {log}_{2} z)

10.T-469 §10.T source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm, simplifying if possible.

$\frac{1}{6} log x -$ $3 log (x + 5)$

Answer

log \frac{\sqrt[6]{x}}{{(x + 5)}^{3}}

10.T-471 §10.T source ↗

In the following exercises, use the Properties of Logarithms to condense the logarithm, simplifying if possible.

Solve for x:
${log}_{7} (x + 2) +$ ${log}_{7} (x - 3) =$ ${log}_{7} 24$

Answer

x = 6

10.T-473 §10.T source ↗

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$5 e^{x - 4} = 40$

Answer

x =

ln 8 +

4 \approx

6.079

10.T-475 §10.T source ↗

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

Researchers recorded that a certain bacteria population grew from 500 to 700 in 5 hours. At this rate of growth, how many bacteria will there be in 20 hours?

Answer

1,921 bacteria

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Problems and answers are adapted from Intermediate Algebra 2e by Lynn Marecek and Andrea Honeycutt Mathis, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.