Deck 5: Exponential and Logarithmic Functions

Question 1

Write the expression as a single logarithm with a coefficient of 1. 2log ₁₀ x - 5log ₁₀ 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) $\log _ { 10 } \frac { 2 x } { 5 y }$

Answer 1

$\log _ { 10 } \frac { x ^ { 2 } } { y ^ { 5 } }$

Question 2

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

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

Answer 2

2

Question 3

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

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) $e = 6 * \sqrt [ 6 ] { e }$

Answer 3

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

Question 4

Simplify the expression by using the definition and properties of logarithms. log ₁₀ 9,000 - log ₁₀ 9

A) 8,991
B) 1
C) 9
D) 3
E) 1,000

Question 5

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. $e ^ { x } - e ^ { - x } = 1$

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 6

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

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 7

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

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

Question 8

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

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 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. $\log _ { 7 } x + \log _ { 7 } ( x + 9 ) = 0$

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 10

Solve the equation. $3 ^ { x } = 243$

A) x = 7
B) x = 5
C) x = 8
D) x = 20
E) no solution

Question 11

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

A)
B)
C)
D)
E)

Question 12

Is the following statement true? (No calculators allowed) $e > \frac { 1 } { 9 }$

Question 13

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

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 14

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.)

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

Question 15

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

Question 16

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

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 17

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.

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) $33.1 ^ { \circ } \mathrm { C } / \text { hour }$

Question 18

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

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

Question 19

Estimate 4 ⁴⁰ in terms of powers of ten.

A) 10 ²²
B) 10 ²⁷
C) 10 ²⁴
D) 10 ¹⁹

Question 20

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

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

Question 21

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.

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

Question 22

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

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 23

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.

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

Question 24

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.

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