Question 1

Solve the inequality. $4 \left( 9 - \left( \frac { 4 } { 35 } \right) ^ { x } \right) \leq 1$

Accepted Answer

A)  $( - \infty , - 1 ]$ 
B)  $( - \infty , - 1 )$ 
C)  $( - 1 , + \infty )$ 
D)  $( - 1 , - 1 ]$ 
E)  $[ - 1 , + \infty )$ 
A)  $( - \infty , - 1 ]$ 
B)  $( - \infty , - 1 )$ 
C)  $( - 1 , + \infty )$ 
D)  $( - 1 , - 1 ]$ 
E)  $[ - 1 , + \infty )$ A

Question 2

Suppose that during the first hour and 15 minutes of a physics experiment, the surface temperature of a small iron block is modeled by the exponential function f(t) = 15e t  , where f(t) is the Celsius temperature t hours after the experiment begins. Compute the average rate of change of temperature over the second half hour of the experiment (i.e., over the interval $0.5 \leq t \leq 1$ ). Round the answer to one decimal place.

Accepted Answer

A)  $32.1 ^ { \circ } \mathrm { C } \text { hour }$ 
B)  $31.8 ^ { \circ } \mathrm { C } / \text { hour }$ 
C)  $41 ^ { \circ } \mathrm { C } / \text { hour }$ 
D)  $32.2 ^ { \circ } \mathrm { C } / \text { hour }$ 
E)  $33.1 ^ { \circ } \mathrm { C } / \text { hour }$ 
A)  $32.1 ^ { \circ } \mathrm { C } \text { hour }$ 
B)  $31.8 ^ { \circ } \mathrm { C } / \text { hour }$ 
C)  $41 ^ { \circ } \mathrm { C } / \text { hour }$ 
D)  $32.2 ^ { \circ } \mathrm { C } / \text { hour }$ 
E)  $33.1 ^ { \circ } \mathrm { C } / \text { hour }$ A

Question 3

A bank offers an interest rate of 4% per annum compounded daily(assuming 365-day year). What is the effective rate?

Accepted Answer

A)  2.88% 
B)  4.01% 
C)  4.28% 
D)  3.04% 
E)  4.08% 
A)  2.88% 
B)  4.01% 
C)  4.28% 
D)  3.04% 
E)  4.08% E

Question 4

Simplify the expression. $\ln e ^ { 2 }$

Accepted Answer

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

Question 5

Graph the pair of functions on the same set of axes. y = 2 x ; y = - 2 x

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 6

Find all the real-number roots of the equation. Give an exact expression for the root and also(where appropriate) a calculator approximation rounded to three decimal places. $\log _ { 7 } x + \log _ { 7 } ( x + 9 ) = 0$

Accepted Answer

A)  $x = \frac { - 9 + \sqrt { 85 } } { 2 } \approx 0.110$ 
B)  $x = \frac { - 9 - \sqrt { 85 } } { 2 } \approx - 9.110$ 
C)  $x = \frac { 9 - \sqrt { 85 } } { 2 } \approx - 0.253$ 
D)  $x = \frac { 9 + \sqrt { 85 } } { 2 } \approx 0.253$ 
E)  no solution 
A)  $x = \frac { - 9 + \sqrt { 85 } } { 2 } \approx 0.110$ 
B)  $x = \frac { - 9 - \sqrt { 85 } } { 2 } \approx - 9.110$ 
C)  $x = \frac { 9 - \sqrt { 85 } } { 2 } \approx - 0.253$ 
D)  $x = \frac { 9 + \sqrt { 85 } } { 2 } \approx 0.253$ 
E)  no solution

Question 7

Solve the equation. Express the answer in terms of natural logarithms. 
5 = 3e 3x - 2

Accepted Answer

A)  $x = \ln 5 - \ln 3 + 2$ 
B)  $x = \frac { \ln 5 - \ln 3 + 2 } { 3 }$ 
C)  $x = \ln 3 + \ln 5 + 2$ 
D)  $x = \frac { \ln 3 + \ln 5 + 2 } { 6 }$ 
E)  $x = \left( \frac { \ln 5 } { 3 } + 2 \right) \div 3$ 
A)  $x = \ln 5 - \ln 3 + 2$ 
B)  $x = \frac { \ln 5 - \ln 3 + 2 } { 3 }$ 
C)  $x = \ln 3 + \ln 5 + 2$ 
D)  $x = \frac { \ln 3 + \ln 5 + 2 } { 6 }$ 
E)  $x = \left( \frac { \ln 5 } { 3 } + 2 \right) \div 3$

Question 8

Write in exponential form. $\log _ { e } \sqrt [ 6 ] { e } = \frac { 1 } { 6 }$

Accepted Answer

A)  $e ^ { \frac { 1 } { 6 } } = \sqrt [ 6 ] { e }$ 
B)  $e ^ { \frac { 1 } { 6 } } = e$ 
C)  $6 ^ { \frac { 1 } { e } } = \sqrt [ 6 ] { e }$ 
D)  $e ^ { \frac { 1 } { 6 } } = \sqrt { 6 }$ 
E)  $e = 6 * \sqrt [ 6 ] { e }$ 
A)  $e ^ { \frac { 1 } { 6 } } = \sqrt [ 6 ] { e }$ 
B)  $e ^ { \frac { 1 } { 6 } } = e$ 
C)  $6 ^ { \frac { 1 } { e } } = \sqrt [ 6 ] { e }$ 
D)  $e ^ { \frac { 1 } { 6 } } = \sqrt { 6 }$ 
E)  $e = 6 * \sqrt [ 6 ] { e }$

Question 9

Is the following statement true? (No calculators allowed) e< $\frac { 5 } { 9 }$

Accepted Answer

A) True 
 B)False

Question 10

You invest $200 at 7% per annum compounded quarterly. How long will it take for your balance to exceed $300? (Round your answer up to the next quarter.)

Accepted Answer

A)  6 quarters 
B)  18 quarters 
C)  24 quarters 
D)  23 quarters 
E)  13 quarters 
A)  6 quarters 
B)  18 quarters 
C)  24 quarters 
D)  23 quarters 
E)  13 quarters

Question 11

Find all the real-number roots of the equation. Give an exact expression for the root and also(where appropriate) a calculator approximation rounded to three decimal places. $e ^ { x } - e ^ { - x } = 1$

Accepted Answer

A)  $x = \pm \ln \frac { - 1 + \sqrt { 5 } } { 2 } \approx \pm 0.479$ 
B)  $x = \ln \frac { 1 + \sqrt { 5 } } { 2 } \approx 0.481$ 
C)  $x = \ln ( 1 + \sqrt { 5 } ) \approx 0.623$ 
D)  $x = \ln \frac { 1 + \sqrt { 5 } } { 2 } \approx 0.481 ; \ln \frac { 1 - \sqrt { 5 } } { 2 } \approx 0.479$ 
E)  no solution 
A)  $x = \pm \ln \frac { - 1 + \sqrt { 5 } } { 2 } \approx \pm 0.479$ 
B)  $x = \ln \frac { 1 + \sqrt { 5 } } { 2 } \approx 0.481$ 
C)  $x = \ln ( 1 + \sqrt { 5 } ) \approx 0.623$ 
D)  $x = \ln \frac { 1 + \sqrt { 5 } } { 2 } \approx 0.481 ; \ln \frac { 1 - \sqrt { 5 } } { 2 } \approx 0.479$ 
E)  no solution

Question 12

Write the expression as a single logarithm with a coefficient of 1. 2log 10 x - 5log 10 y

Accepted Answer

A)  $- 3 \log _ { 10 } \left( \frac { x } { y } \right)$ 
B)  $\log _ { 10 } ( 2 x - 5 y )$ 
C)  $\log _ { 10 } \left( x ^ { 2 } - y ^ { 5 } \right)$ 
D)  $\log _ { 10 } \frac { x ^ { 2 } } { y ^ { 5 } }$ 
E)  $\log _ { 10 } \frac { 2 x } { 5 y }$ 
A)  $- 3 \log _ { 10 } \left( \frac { x } { y } \right)$ 
B)  $\log _ { 10 } ( 2 x - 5 y )$ 
C)  $\log _ { 10 } \left( x ^ { 2 } - y ^ { 5 } \right)$ 
D)  $\log _ { 10 } \frac { x ^ { 2 } } { y ^ { 5 } }$ 
E)  $\log _ { 10 } \frac { 2 x } { 5 y }$

Question 13

The radioactive isotope carbone-14 is used as a tracer in medical and biological research. Compute the half-life of carbone-14 given that the decay constant k is $- 1.2097 \times 10 ^ { - 4 }$ (The units for k here are such that your half-life answer will be in years.) Please round the answer to one decimal place.

Accepted Answer

A)  5,728.7years 
B)  5,729.2years 
C)  5,758.5years 
D)  5,730.3years 
E)  5,729.9years 
A)  5,728.7years 
B)  5,729.2years 
C)  5,758.5years 
D)  5,730.3years 
E)  5,729.9years

Question 14

You place a sum of $600 in a savings account at 4% per annum compounded continuously. When will the balance reach $800?

Accepted Answer

A)  in 7.23 years 
B)  in 5.99 years 
C)  in 15.59 years 
D)  in 7.19 years 
E)  in 4.6 years 
A)  in 7.23 years 
B)  in 5.99 years 
C)  in 15.59 years 
D)  in 7.19 years 
E)  in 4.6 years

Question 15

Use properties of exponents to simplify. $$\left( 9 ^ { 1 + \sqrt { 7 } } \right) \left( 9 ^ { 1 - \sqrt { 7 } } \right)$$

Accepted Answer

A)  49 
B)  81 
C)  16 
D)  $9 \sqrt { 7 }$ 
E)  343 
A)  49 
B)  81 
C)  16 
D)  $9 \sqrt { 7 }$ 
E)  343

Question 16

The functions $f ( x ) = 3 ^ { x }$ and $g ( x )$ are inverse functions. Find the $g ( x )$

Accepted Answer

A)  $g ( x ) = \log _ { 3 } x$ 
B)  $g ( x ) = \ln x$ 
C)  $g ( x ) = \sqrt [ 3 ] { x }$ 
D)  $g ( x ) = x ^ { 3 }$ 
E)  $g ( x ) = \log x$ 
A)  $g ( x ) = \log _ { 3 } x$ 
B)  $g ( x ) = \ln x$ 
C)  $g ( x ) = \sqrt [ 3 ] { x }$ 
D)  $g ( x ) = x ^ { 3 }$ 
E)  $g ( x ) = \log x$

Question 17

The half-life of radium-226 is 1620 years. How much of an initial 2-g sample remains after 500 years? Please round the answer to two decimal places.

Accepted Answer

A)  0.81 grams 
B)  1.54 grams 
C)  1.61 grams 
D)  1.63 grams 
E)  1.65 grams 
A)  0.81 grams 
B)  1.54 grams 
C)  1.61 grams 
D)  1.63 grams 
E)  1.65 grams

Question 18

Find all the real-number roots of the equations. $\ln \left( x ^ { 11 } \right) = 11 \ln x$ $( \ln y ) ^ { 11 } = 11 \ln y$

Accepted Answer

A)  $x > 0 ; y = 1 , y = \frac { e ^ { 11 } } { 10 }$ 
B)  $x = 1 ; y = e ^ { \pm \sqrt [ 10 ] { 11 } }$ 
C)  $x > 0 ; y = e ^ { \pm \sqrt [ 11 ] { 10 } }$ 
D)  $x = 1 ; y = 1 , y = e ^ { \sqrt [ 11 ] { 10 } }$ 
E)  $x > 0 ; y = 1 , y = e ^ { \pm \sqrt [ 10 ] { 11 } }$ 
A)  $x > 0 ; y = 1 , y = \frac { e ^ { 11 } } { 10 }$ 
B)  $x = 1 ; y = e ^ { \pm \sqrt [ 10 ] { 11 } }$ 
C)  $x > 0 ; y = e ^ { \pm \sqrt [ 11 ] { 10 } }$ 
D)  $x = 1 ; y = 1 , y = e ^ { \sqrt [ 11 ] { 10 } }$ 
E)  $x > 0 ; y = 1 , y = e ^ { \pm \sqrt [ 10 ] { 11 } }$

Question 19

Complete the table. $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & & & & & & & \
\hline
\end{array}$$

Accepted Answer

A)  $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & 1 & 3 & 9 & 27 & - 27 & - 9 & - 3 \
\hline
\end{array}$$ 
B)  $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & 0 & 1 & 2 & 3 & 1 & 2 & 3 \
\hline
\end{array}$$ 
C)  $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & - 1 & - 3 & - 9 & - 27 & 27 & 9 & 3 \
\hline
\end{array}$$ 
D)  $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & 0 & 1 & 2 & 3 & - 1 & - 2 & - 3 \
\hline
\end{array}$$ 
E)  $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & \frac { 1 } { 3 } & 1 & 3 & 9 & \frac { 1 } { 9 } & \frac { 1 } { 27 } & \frac { 1 } { 81 } \
\hline
\end{array}$$ 
A)  $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & 1 & 3 & 9 & 27 & - 27 & - 9 & - 3 \
\hline
\end{array}$$ 
B)  $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & 0 & 1 & 2 & 3 & 1 & 2 & 3 \
\hline
\end{array}$$ 
C)  $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & - 1 & - 3 & - 9 & - 27 & 27 & 9 & 3 \
\hline
\end{array}$$ 
D)  $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & 0 & 1 & 2 & 3 & - 1 & - 2 & - 3 \
\hline
\end{array}$$ 
E)  $$\begin{array} { | c | c | c | c | c | c | c | c | } 
\hline x & 1 & 3 & 3 ^ { 2 } & 3 ^ { 3 } & 3 ^ { - 1 } & 3 ^ { - 2 } & 3 ^ { - 3 } \
\hline \log 3 x & \frac { 1 } { 3 } & 1 & 3 & 9 & \frac { 1 } { 9 } & \frac { 1 } { 27 } & \frac { 1 } { 81 } \
\hline
\end{array}$$

Question 20

In 2000, the nations of Niger and Portugal had similar size populations: Niger 10.1 million, Portugal 10.0 million. However, the relative growth rate for Niger was 3.0%/year, whereas that for Portugal was only 0.1%/year. Assuming exponential growth at the given rates, make projections for each population in the year 2055. Please round the answers to one decimal place.

Accepted Answer

A)  Niger: 10.6 million; Portugal: 52.6 million 
B)  Niger: 53.8 million; Portugal: 11.0 million 
C)  Niger: 52.6 million; Portugal: 10.6 million 
D)  Niger: 54.4 million; Portugal: 11.6 million 
E)  Niger: 54.0 million; Portugal: 11.1 million 
A)  Niger: 10.6 million; Portugal: 52.6 million 
B)  Niger: 53.8 million; Portugal: 11.0 million 
C)  Niger: 52.6 million; Portugal: 10.6 million 
D)  Niger: 54.4 million; Portugal: 11.6 million 
E)  Niger: 54.0 million; Portugal: 11.1 million

Solve the inequality. $4 \left( 9 - \left( \frac { 4 } { 35 } \right) ^ { x } \right) \leq 1$

A bank offers an interest rate of 4% per annum compounded daily(assuming 365-day year). What is the effective rate?

Simplify the expression. $\ln e ^ { 2 }$

Graph the pair of functions on the same set of axes. y = 2^x; y = - 2^x

Find all the real-number roots of the equation. Give an exact expression for the root and also(where appropriate) a calculator approximation rounded to three decimal places. $\log _ { 7 } x + \log _ { 7 } ( x + 9 ) = 0$

Solve the equation. Express the answer in terms of natural logarithms. 5 = 3e³^x^{- 2}

Write in exponential form. $\log _ { e } \sqrt [ 6 ] { e } = \frac { 1 } { 6 }$

Is the following statement true? (No calculators allowed) e< $\frac { 5 } { 9 }$

You invest $200 at 7% per annum compounded quarterly. How long will it take for your balance to exceed $300? (Round your answer up to the next quarter.)

Find all the real-number roots of the equation. Give an exact expression for the root and also(where appropriate) a calculator approximation rounded to three decimal places. $e ^ { x } - e ^ { - x } = 1$

Write the expression as a single logarithm with a coefficient of 1. 2log₁₀x - 5log₁₀y

You place a sum of $600 in a savings account at 4% per annum compounded continuously. When will the balance reach $800?

Use properties of exponents to simplify. $\left( 9 ^ { 1 + \sqrt { 7 } } \right) \left( 9 ^ { 1 - \sqrt { 7 } } \right)$

The functions $f ( x ) = 3 ^ { x }$ and $g ( x )$ are inverse functions. Find the $g ( x )$

The half-life of radium-226 is 1620 years. How much of an initial 2-g sample remains after 500 years? Please round the answer to two decimal places.

Find all the real-number roots of the equations. $\ln \left( x ^ { 11 } \right) = 11 \ln x$ $( \ln y ) ^ { 11 } = 11 \ln y$

Complete the table. x 1 3 3x

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Graphs of the Trigonometric Functions

Analytical Trigonometry

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Roots of Polynomial Equations

Additional Topics in Algebra

Filters

Exam 5: Exponential and Logarithmic Functions

Solve the inequality. 4(9−(435)x)≤14 \left( 9 - \left( \frac { 4 } { 35 } \right) ^ { x } \right) \leq 14(9−(354​)x)≤1

A bank offers an interest rate of 4% per annum compounded daily(assuming 365-day year). What is the effective rate?

Simplify the expression. ln⁡e2\ln e ^ { 2 }lne2

Graph the pair of functions on the same set of axes. y = 2 x ; y = - 2 x

Find all the real-number roots of the equation. Give an exact expression for the root and also(where appropriate) a calculator approximation rounded to three decimal places. log⁡7x+log⁡7(x+9)=0\log _ { 7 } x + \log _ { 7 } ( x + 9 ) = 0log7​x+log7​(x+9)=0

Solve the equation. Express the answer in terms of natural logarithms. 5 = 3e 3x - 2

Write in exponential form. log⁡ee6=16\log _ { e } \sqrt [ 6 ] { e } = \frac { 1 } { 6 }loge​6e​=61​

Is the following statement true? (No calculators allowed) e< 59\frac { 5 } { 9 }95​

You invest $200 at 7% per annum compounded quarterly. How long will it take for your balance to exceed $300? (Round your answer up to the next quarter.)

Find all the real-number roots of the equation. Give an exact expression for the root and also(where appropriate) a calculator approximation rounded to three decimal places. ex−e−x=1e ^ { x } - e ^ { - x } = 1ex−e−x=1

Write the expression as a single logarithm with a coefficient of 1. 2log 10 x - 5log 10 y

You place a sum of $600 in a savings account at 4% per annum compounded continuously. When will the balance reach $800?

Use properties of exponents to simplify. (91+7)(91−7)\left( 9 ^ { 1 + \sqrt { 7 } } \right) \left( 9 ^ { 1 - \sqrt { 7 } } \right)(91+7​)(91−7​)

The functions f(x)=3xf ( x ) = 3 ^ { x }f(x)=3x and g(x)g ( x )g(x) are inverse functions. Find the g(x)g ( x )g(x)

The half-life of radium-226 is 1620 years. How much of an initial 2-g sample remains after 500 years? Please round the answer to two decimal places.

Find all the real-number roots of the equations. ln⁡(x11)=11ln⁡x\ln \left( x ^ { 11 } \right) = 11 \ln xln(x11)=11lnx (ln⁡y)11=11ln⁡y( \ln y ) ^ { 11 } = 11 \ln y(lny)11=11lny

Complete the table. x 1 3 3x

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Graphs of the Trigonometric Functions

Analytical Trigonometry

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Roots of Polynomial Equations

Additional Topics in Algebra

Filters

Solve the inequality. $4 \left( 9 - \left( \frac { 4 } { 35 } \right) ^ { x } \right) \leq 1$

Simplify the expression. $\ln e ^ { 2 }$

Graph the pair of functions on the same set of axes. y = 2^x; y = - 2^x

Find all the real-number roots of the equation. Give an exact expression for the root and also(where appropriate) a calculator approximation rounded to three decimal places. $\log _ { 7 } x + \log _ { 7 } ( x + 9 ) = 0$

Solve the equation. Express the answer in terms of natural logarithms. 5 = 3e³^x^{- 2}

Write in exponential form. $\log _ { e } \sqrt [ 6 ] { e } = \frac { 1 } { 6 }$

Is the following statement true? (No calculators allowed) e< $\frac { 5 } { 9 }$

Find all the real-number roots of the equation. Give an exact expression for the root and also(where appropriate) a calculator approximation rounded to three decimal places. $e ^ { x } - e ^ { - x } = 1$

Write the expression as a single logarithm with a coefficient of 1. 2log₁₀x - 5log₁₀y

Use properties of exponents to simplify. $\left( 9 ^ { 1 + \sqrt { 7 } } \right) \left( 9 ^ { 1 - \sqrt { 7 } } \right)$

The functions $f ( x ) = 3 ^ { x }$ and $g ( x )$ are inverse functions. Find the $g ( x )$

Find all the real-number roots of the equations. $\ln \left( x ^ { 11 } \right) = 11 \ln x$ $( \ln y ) ^ { 11 } = 11 \ln y$