Question 1

Which of the following functions is one-to-one?

Accepted Answer

$f(x) = 2^x$

Question 2

Given $f(x) = 3x + 2$ and $g(x) = \frac{x-2}{3}$, which statement is true?

Accepted Answer

$f$ and $g$ are inverses because $f(g(x)) = x$ and $g(f(x)) = x$

Question 3

If $f(x) = 2^x$ and $g(x) = \log_2 x$, what is $f(g(8))$?

Accepted Answer

$8$

Question 4

Which of the following is the correct inverse of $f(x) = e^{2x}$?

Accepted Answer

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

Question 5

Given the exponential function $f(x) = 5 \cdot 3^x$, which of the following is a correct table of values for $x = 0, 1, 2$?

Accepted Answer

$x=0: 5,\ x=1: 15,\ x=2: 45$

Question 6

Solve the logarithmic equation: $\log_2(x) = 5$.

Accepted Answer

$x = 32$

Question 7

Convert the logarithmic equation $\log_{10}(x) = 3$ to its equivalent exponential form.

Accepted Answer

$x = 10^3$

Question 8

Convert the exponential equation $4^x = 20$ to a logarithmic equation.

Accepted Answer

$x = \log_4 20$

Question 9

Find the value of $\log_5 125$ using the change of base formula.

Accepted Answer

$\frac{\ln 125}{\ln 5}$

Question 10

Express $\log_3(9x)$ as a sum using logarithmic properties.

Accepted Answer

$\log_3 9 + \log_3 x$

Question 11

Express $\log_2 32 - \log_2 4$ as a single logarithm.

Accepted Answer

$\log_2 8$

Question 12

Expand $\log_5 \left(\frac{a^3 b}{c^2}\right)$ using logarithmic properties.

Accepted Answer

$3\log_5 a + \log_5 b - 2\log_5 c$

Question 13

Solve for $x$: $2^{x+1} = 16$.

Accepted Answer

$x = 3$

Question 14

Solve for $x$: $\log_3(x-1) = 2$.

Accepted Answer

$x = 10$

Question 15

Solve the exponential equation $5^{2x} = 125$ by taking the log of both sides.

Accepted Answer

$x = 1.5$

Question 16

A $\$1000$ investment grows at an annual rate of $6\%$ compounded quarterly. What is the amount after $5$ years? Use $A(t) = P \left[1 + \frac{r}{n}\right]^{nt}$.

Accepted Answer

$A = 1000 \left[1 + \frac{0.06}{4}\right]^{20}$

Question 17

A population of bacteria grows continuously at a rate of $8\%$ per hour. If the initial population is $500$, what is the population after $10$ hours? Use $A(t) = Pe^{rt}$.

Accepted Answer

$A = 500e^{0.08 	imes 10}$

Question 18

Which of the following is equivalent to $\log_7 49$?

Accepted Answer

$2$

Question 19

If $\log_b 2 = 0.3010$ and $\log_b 5 = 0.6990$, what is $\log_b 20$?

Accepted Answer

$1.0000$

Question 20

Which of the following best describes the graph of $y = \log_2 x$?

Accepted Answer

It passes through $(1,0)$ and increases slowly for large $x$.

Exponential and Logarithmic Functions: College Algebra Study Guide

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