Question 1

How long would it take $\$ 9000$ to grow to $\$ 36,000$ at $5 \%$ compounded continuously? Round your answer to the nearest tenth of a year.

Accepted Answer

A)  28.0 years 
B)  28.6 years 
C)  35.0 years 
D)  27.7 years 
A)  28.0 years 
B)  28.6 years 
C)  35.0 years 
D)  27.7 years D

Question 2

If the following defines a one-to-one function, find its inverse. If not, write "Not one-to-one."
-$\left\{\left(-3, \frac{1}{2}\right),(-2,-3),\left(-\frac{1}{3}, 2\right),\left(-9, \frac{3}{8}\right)\right\}$

Accepted Answer

A)  $\left\{\left(\frac{1}{2}, 3\right),(-3,2),\left(2, \frac{1}{3}\right),\left(-\frac{3}{8},-9\right)\right\}$ 
B) $\left\{\left(\frac{1}{3},-3\right),(-3,-2),\left(2,-\frac{1}{2}\right),\left(\frac{3}{8},-9\right)\right\}$ 
C)  Not one-to-one 
D)  $\left\{\left(\frac{1}{2},-3\right),(-3,-2),\left(2,-\frac{1}{3}\right),\left(\frac{3}{8},-9\right)\right\}$ 
A)  $\left\{\left(\frac{1}{2}, 3\right),(-3,2),\left(2, \frac{1}{3}\right),\left(-\frac{3}{8},-9\right)\right\}$ 
B) $\left\{\left(\frac{1}{3},-3\right),(-3,-2),\left(2,-\frac{1}{2}\right),\left(\frac{3}{8},-9\right)\right\}$ 
C)  Not one-to-one 
D)  $\left\{\left(\frac{1}{2},-3\right),(-3,-2),\left(2,-\frac{1}{3}\right),\left(\frac{3}{8},-9\right)\right\}$ D

Question 3

Solve the equation. Use natural logarithms. When appropriate, give solutions to three decimal places unless otherwise indicated.
-$\ln \mathrm{e}^{4 \mathrm{x}}=20$

Accepted Answer

A)  $\{80\}$ 
B)  $\{16\}$ 
C)  $\{24\}$ 
D)  $\{5\}$ 
A)  $\{80\}$ 
B)  $\{16\}$ 
C)  $\{24\}$ 
D)  $\{5\}$ D

Question 4

The population of a small country increases according to the function $B=2,500,000 e^{0.05 t}$, where $t$ is measured in years. How many people will the country have after 9 years?

Accepted Answer

A)  $3,920,780$ 
B)  $7,045,957$ 
C)  866,969 
D)  $1,996,269$ 
A)  $3,920,780$ 
B)  $7,045,957$ 
C)  866,969 
D)  $1,996,269$

Question 5

Using the exponential key of a calculator to find an approximation to the nearest thousandth.
-$0.4^{3.725}$

Accepted Answer

A)  192.533 
B)  0.033 
C)  1.49 
D)  1.6 
A)  192.533 
B)  0.033 
C)  1.49 
D)  1.6

Question 6

A sample of 700 grams of radioactive substance decays according to the function $A(t)=700 e^{-0.034 t}$, where $t$ is the time in years. How much of the substance will be left in the sample after 30 years? Round your answer to the nearest whole gram.

Accepted Answer

A)  $2213 \mathrm{~g}$ 
B)  $0 \mathrm{~g}$ 
C)  $1 \mathrm{~g}$ 
D)  $252 \mathrm{~g}$ 
A)  $2213 \mathrm{~g}$ 
B)  $0 \mathrm{~g}$ 
C)  $1 \mathrm{~g}$ 
D)  $252 \mathrm{~g}$

Question 7

Use the change-of-base rule to express the given logarithm in terms of common logarithms, in terms of natural logarithms, and correct to four decimal places.
-$\log _{27} 93.10$
$\log 27$

Accepted Answer

A)  $\frac{\log 93.10}{\log 27} ; \frac{\ln 93.10}{\ln 27} ; 1.3756$ 
B)  $\frac{\log 27}{\log 93.10} ; \frac{\ln 27}{\ln 93.10} ; 0.7270$ 
C)  $\frac{\log 93.10}{\log 10} ; \frac{\ln 93.10}{\ln 10} ; 1.9689$ 
D)  $\frac{93.10}{27} ; 3.4481$ 
A)  $\frac{\log 93.10}{\log 27} ; \frac{\ln 93.10}{\ln 27} ; 1.3756$ 
B)  $\frac{\log 27}{\log 93.10} ; \frac{\ln 27}{\ln 93.10} ; 0.7270$ 
C)  $\frac{\log 93.10}{\log 10} ; \frac{\ln 93.10}{\ln 10} ; 1.9689$ 
D)  $\frac{93.10}{27} ; 3.4481$

Question 8

The function $\mathrm{A}=\mathrm{A}_{0} \mathrm{e}^{-0.0099 \mathrm{x}}$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $\mathrm{x}$ is the number of years since the material was put into the vault. If 200 pounds of the material are initially put into the vault, how many pounds will be left after 160 years?

Accepted Answer

A)  $44 \mathrm{lb}$ 
B)  $229 \mathrm{lb}$ 
C)  $41 \mathrm{lb}$ 
D)  $148 \mathrm{lb}$ 
A)  $44 \mathrm{lb}$ 
B)  $229 \mathrm{lb}$ 
C)  $41 \mathrm{lb}$ 
D)  $148 \mathrm{lb}$

Question 9

With the exponential function $f(x)=a^{x}$, why must $a \neq 0$ ?

Accepted Answer

Answers will vary. (If allowed

Question 10

Find the indicated value.
-Let $f(x)=2^{x} \cdot f^{-1}\left(\frac{1}{16}\right)$

Accepted Answer

A)  -2 
B)  -4 
C)  4 
D)  $-\frac{1}{16}$ 
A)  -2 
B)  -4 
C)  4 
D)  $-\frac{1}{16}$

Question 11

Solve the equation. Give the exact solution or solutions.
-$\log _{3}(5 x+7)=\log _{3}(5 x+4)$

Accepted Answer

A)  $\{0\}$ 
B)  $\{3\}$ 
C)  $\varnothing$ 
D)  $\{5\}$ 
A)  $\{0\}$ 
B)  $\{3\}$ 
C)  $\varnothing$ 
D)  $\{5\}$

Question 12

An animal species is introduced into a certain area. Its population is approximated by $F(t)=400 \log 10(2 t+3)$, where $t$ represents the number of months since its introduction. Find the population of this species 6 months after its introduction into the area. Round answer to the nearest whole number.

Accepted Answer

A)  704 animals 
B)  74 animals 
C)  240 animals 
D)  470 animals 
A)  704 animals 
B)  74 animals 
C)  240 animals 
D)  470 animals

Question 13

What is the range of the function $y=\log _{7} x$ ?

Accepted Answer

A)  $[0, \infty)$ 
B)  $(7, \infty)$ 
C)  $(0, \infty)$ 
D)  $(\infty, \infty)$ 
A)  $[0, \infty)$ 
B)  $(7, \infty)$ 
C)  $(0, \infty)$ 
D)  $(\infty, \infty)$

Question 14

Determine whether or not the function is one-to-one.
-$f(x)=x^{2}-5$

Accepted Answer

A) True 
 B)False

Question 15

Solve the equation.
-$\log _{4}(x+4)+\log _{4}(x-2)=2$

Accepted Answer

A)  $x=4$ 
B)  $x=4, x=-6$ 
C)  $x=5$ 
D)  $x=-6$ 
A)  $x=4$ 
B)  $x=4, x=-6$ 
C)  $x=5$ 
D)  $x=-6$

Question 16

Write in exponential form.
-$\log _{10} 0.0001=-4$

Accepted Answer

A)  $10^{4}=0.1$ 
B)  $10^{-4}=0.0001$ 
C)  $10^{4}=0.0001$ 
D)  $10^{-4}=0.1$ 
A)  $10^{4}=0.1$ 
B)  $10^{-4}=0.0001$ 
C)  $10^{4}=0.0001$ 
D)  $10^{-4}=0.1$

Question 17

$\frac{\log _{8} 10}{\log _{8} 80}=\frac{1}{8}$

Accepted Answer

A) True 
 B)False

Question 18

Write in exponential form.
-$\log _{4} \frac{1}{64}=x$

Accepted Answer

A)  $\left\{\frac{1}{256}\right\}$ 
B)  $\left\{\frac{1}{16}\right\}$ 
C)  $\{-3\}$ 
D)  $\{3\}$ 
A)  $\left\{\frac{1}{256}\right\}$ 
B)  $\left\{\frac{1}{16}\right\}$ 
C)  $\{-3\}$ 
D)  $\{3\}$

Question 19

The half-life of a certain radioactive substance is 36 years. Suppose that at time $t=0$, there are $27 \mathrm{~g}$ of the substance. Then after $t$ years, the number of grams of the substance remaining will be $\mathrm{N}(\mathrm{t})=$ $27(1 / 2)^{t / 36}$. How many grams of the substance will remain after 198 years?

Accepted Answer

A)  $0.6 \mathrm{~g}$ 
B)  $0.3 \mathrm{~g}$ 
C)  $0.07 \mathrm{~g}$ 
D)  $0.15 \mathrm{~g}$ 
A)  $0.6 \mathrm{~g}$ 
B)  $0.3 \mathrm{~g}$ 
C)  $0.07 \mathrm{~g}$ 
D)  $0.15 \mathrm{~g}$

Question 20

Write in exponential form.
-$\log _{4} \sqrt{4^{3}}=x$

Accepted Answer

A)  $\left\{\frac{3}{2}\right\}$ 
B)  $\{3\}$ 
C)  $\{\sqrt{3}\}$ 
D)  $\{12\}$ 
A)  $\left\{\frac{3}{2}\right\}$ 
B)  $\{3\}$ 
C)  $\{\sqrt{3}\}$ 
D)  $\{12\}$

How long would it take $\$ 9000$ to grow to $\$ 36,000$ at $5 \%$ compounded continuously? Round your answer to the nearest tenth of a year.

If the following defines a one-to-one function, find its inverse. If not, write "Not one-to-one." - $\left\{\left(-3, \frac{1}{2}\right),(-2,-3),\left(-\frac{1}{3}, 2\right),\left(-9, \frac{3}{8}\right)\right\}$

Solve the equation. Use natural logarithms. When appropriate, give solutions to three decimal places unless otherwise indicated. - $\ln \mathrm{e}^{4 \mathrm{x}}=20$

The population of a small country increases according to the function $B=2,500,000 e^{0.05 t}$ , where $t$ is measured in years. How many people will the country have after 9 years?

Using the exponential key of a calculator to find an approximation to the nearest thousandth. - $0.4^{3.725}$

A sample of 700 grams of radioactive substance decays according to the function $A(t)=700 e^{-0.034 t}$ , where $t$ is the time in years. How much of the substance will be left in the sample after 30 years? Round your answer to the nearest whole gram.

Use the change-of-base rule to express the given logarithm in terms of common logarithms, in terms of natural logarithms, and correct to four decimal places. - $\log _{27} 93.10$ $\log 27$

With the exponential function $f(x)=a^{x}$ , why must $a \neq 0$ ?

Find the indicated value. -Let $f(x)=2^{x} \cdot f^{-1}\left(\frac{1}{16}\right)$

Solve the equation. Give the exact solution or solutions. - $\log _{3}(5 x+7)=\log _{3}(5 x+4)$

What is the range of the function $y=\log _{7} x$ ?

Determine whether or not the function is one-to-one. - $f(x)=x^{2}-5$

Solve the equation. - $\log _{4}(x+4)+\log _{4}(x-2)=2$

Write in exponential form. - $\log _{10} 0.0001=-4$

$\frac{\log _{8} 10}{\log _{8} 80}=\frac{1}{8}$

Write in exponential form. - $\log _{4} \frac{1}{64}=x$

Write in exponential form. - $\log _{4} \sqrt{4^{3}}=x$

Review of the Real Number System

Linear Equations, Inequalities, and Applications

Linear Equations, Graphs, and Functions

Systems of Linear Equations

Exponents, Polynomials, and Polynomial Functions

Factoring

Rational Expressions and Functions

Roots, Radicals, and Root Functions

Quadratic Equations, Inequalities, and Functions

Nonlinear Functions, Conic Sections, and Nonlinear Systems

Further Topics in Algebra

Appendices

Filters

Exam 10: Inverse, Exponential, and Logarithmic Functions

How long would it take $9000\$ 9000$9000 to grow to $36,000\$ 36,000$36,000 at 5%5 \%5% compounded continuously? Round your answer to the nearest tenth of a year.

Solve the equation. Use natural logarithms. When appropriate, give solutions to three decimal places unless otherwise indicated. - ln⁡e4x=20\ln \mathrm{e}^{4 \mathrm{x}}=20lne4x=20

The population of a small country increases according to the function B=2,500,000e0.05tB=2,500,000 e^{0.05 t}B=2,500,000e0.05t , where ttt is measured in years. How many people will the country have after 9 years?

Using the exponential key of a calculator to find an approximation to the nearest thousandth. - 0.43.7250.4^{3.725}0.43.725

A sample of 700 grams of radioactive substance decays according to the function A(t)=700e−0.034tA(t)=700 e^{-0.034 t}A(t)=700e−0.034t , where ttt is the time in years. How much of the substance will be left in the sample after 30 years? Round your answer to the nearest whole gram.

Use the change-of-base rule to express the given logarithm in terms of common logarithms, in terms of natural logarithms, and correct to four decimal places. - log⁡2793.10\log _{27} 93.10log27​93.10 log⁡27\log 27log27

With the exponential function f(x)=axf(x)=a^{x}f(x)=ax , why must a≠0a \neq 0a=0 ?

Find the indicated value. -Let f(x)=2x⋅f−1(116)f(x)=2^{x} \cdot f^{-1}\left(\frac{1}{16}\right)f(x)=2x⋅f−1(161​)

Solve the equation. Give the exact solution or solutions. - log⁡3(5x+7)=log⁡3(5x+4)\log _{3}(5 x+7)=\log _{3}(5 x+4)log3​(5x+7)=log3​(5x+4)

What is the range of the function y=log⁡7xy=\log _{7} xy=log7​x ?

Determine whether or not the function is one-to-one. - f(x)=x2−5f(x)=x^{2}-5f(x)=x2−5

Solve the equation. - log⁡4(x+4)+log⁡4(x−2)=2\log _{4}(x+4)+\log _{4}(x-2)=2log4​(x+4)+log4​(x−2)=2

Write in exponential form. - log⁡100.0001=−4\log _{10} 0.0001=-4log10​0.0001=−4

log⁡810log⁡880=18\frac{\log _{8} 10}{\log _{8} 80}=\frac{1}{8}log8​80log8​10​=81​

Write in exponential form. - log⁡4164=x\log _{4} \frac{1}{64}=xlog4​641​=x

Write in exponential form. - log⁡443=x\log _{4} \sqrt{4^{3}}=xlog4​43​=x

Review of the Real Number System

Linear Equations, Inequalities, and Applications

Linear Equations, Graphs, and Functions

Systems of Linear Equations

Exponents, Polynomials, and Polynomial Functions

Factoring

Rational Expressions and Functions

Roots, Radicals, and Root Functions

Quadratic Equations, Inequalities, and Functions

Nonlinear Functions, Conic Sections, and Nonlinear Systems

Further Topics in Algebra

Appendices

Filters

How long would it take $\$ 9000$ to grow to $\$ 36,000$ at $5 \%$ compounded continuously? Round your answer to the nearest tenth of a year.

Solve the equation. Use natural logarithms. When appropriate, give solutions to three decimal places unless otherwise indicated. - $\ln \mathrm{e}^{4 \mathrm{x}}=20$

The population of a small country increases according to the function $B=2,500,000 e^{0.05 t}$ , where $t$ is measured in years. How many people will the country have after 9 years?

Using the exponential key of a calculator to find an approximation to the nearest thousandth. - $0.4^{3.725}$

A sample of 700 grams of radioactive substance decays according to the function $A(t)=700 e^{-0.034 t}$ , where $t$ is the time in years. How much of the substance will be left in the sample after 30 years? Round your answer to the nearest whole gram.

Use the change-of-base rule to express the given logarithm in terms of common logarithms, in terms of natural logarithms, and correct to four decimal places. - $\log _{27} 93.10$ $\log 27$

With the exponential function $f(x)=a^{x}$ , why must $a \neq 0$ ?

Find the indicated value. -Let $f(x)=2^{x} \cdot f^{-1}\left(\frac{1}{16}\right)$

Solve the equation. Give the exact solution or solutions. - $\log _{3}(5 x+7)=\log _{3}(5 x+4)$

What is the range of the function $y=\log _{7} x$ ?

Determine whether or not the function is one-to-one. - $f(x)=x^{2}-5$

Solve the equation. - $\log _{4}(x+4)+\log _{4}(x-2)=2$

Write in exponential form. - $\log _{10} 0.0001=-4$

$\frac{\log _{8} 10}{\log _{8} 80}=\frac{1}{8}$

Write in exponential form. - $\log _{4} \frac{1}{64}=x$

Write in exponential form. - $\log _{4} \sqrt{4^{3}}=x$