Deck 4: Exponential and Logarithmic Functions

Question 1

Evaluate the given expression. $$\left( \frac { 625 } { 1,296 } \right) ^ { 5 / 4 }$$

A) $\frac { 3,125 } { 1,296 }$
B) $\frac { 3,125 } { 7,776 }$
C) $\frac { 62.5 } { 1,296 }$
D) $\frac { 5 } { 6 }$

Answer 1

$\frac { 3,125 } { 7,776 }$

Question 2

Simplify $64 ^ { 2 / 3 }$ .

Answer 2

16

Question 3

Let $f ( x ) = 6 x ^ { 5 } - 90 \ln x$ , for x > 0. Find the minimum value of f for x > 0.

A) 0
B) $3 \left( 3 ^ { 5 } - 15 \ln 3 \right)$
C) 18(1 - ln 3)
D) $6 \left( 3 ^ { 5 } - 15 \ln 3 \right)$

Answer 3

18(1 - ln 3)

Question 4

Simplify $\left( 16 ^ { 1 / 2 } - 4 ^ { 1 / 2 } \right) ^ { - 2 }$ .

Question 5

Simplify $\left( 49 ^ { 1 / 2 } - 9 ^ { 1 / 2 } \right) ^ { - 2 }$ .

Question 6

Simplify $\frac { 3 ^ { 5 } } { 3 ^ { 4 } }$ .

A) 3
B) $\frac { 1 } { 3 }$
C) 9
D) $\frac { 1 } { 9 }$

Question 7

Simplify $\frac { 9 ^ { 3 } } { 3 ^ { 2 } }$ .

A) $\frac { 1 } { 81 }$
B) 81
C) $\frac { 1 } { 6,561 }$
D) 6,561

Question 8

Simplify: $\frac { 4 ^ { 5 } } { 4 ^ { 4 } }$

A) 4
B) $\frac { 1 } { 4 }$
C) $4 ^ { 9 }$
D) $\frac { 1 } { 16 }$

Question 9

$\frac { 3 ^ { 4 } } { 3 ^ { 6 } } = \frac { 1 } { 9 }$

Question 10

$\frac { 8 ^ { 6 } } { 8 ^ { 9 } } = \frac { 1 } { 512 }$

Question 11

Simplify: $\left( 81 x ^ { 7 } y ^ { 10 } \right) ^ { 1 / 2 }$

A) $9 x ^ { 7 / 2 } y ^ { 5 }$
B) $\frac { 81 } { 2 } x ^ { 7 / 2 } y ^ { 5 }$
C) $81 x ^ { 13 / 2 } y ^ { 10 / 2 }$
D) $9 x ^ { 5 } y ^ { 8 }$

Question 12

Simplify. $\left( 2 x ^ { 9 } y ^ { - 8 } \right) \left( 4 x ^ { - 12 } y ^ { 9 } \right)$ Your answer should include only positive exponents.

Question 13

Simplify. $\left( 4 x ^ { 2 } y ^ { - 6 } z ^ { 3 } \right) ^ { - 2 }$ Your answer should include only positive exponents.

Question 14

Find all real numbers x that satisfy the given equation. 3^x2^2x = 1,728

A) 6
B) 3
C) 9
D) 12

Question 15

Find all real numbers x that satisfy the given equation. $\left( \frac { 1 } { 9 } \right) ^ { x - 1 } = 3 ^ { 2 + 5 x ^ { 2 } }$

A) 0
B) $\frac { 2 } { 5 }$
C) $0 , \frac { 2 } { 5 }$
D) $0 , - \frac { 2 } { 5 }$

Question 16

Sketch the graph of $f ( x ) = 3 - e ^ { x }$ .

Question 17

Suppose $1,500 is invested at an annual interest rate of 8 percent compounded quarterly. Compute the balance after 12 years.

A) $3,780.61
B) $3,820.61
C) $3,880.61
D) $3,890.61

Question 18

How quickly will $2,000 grow to $10,000 if interest is 7% compounded quarterly? Round your answer to the nearest whole year, if necessary.

A) about 20 years
B) about 21 years
C) about 22 years
D) about 23 years

Question 19

How quickly will $700 grow to $2,000 if interest is 8% compounded quarterly? Round your answer to two decimal places, if necessary.

A) in 13.00 years
B) in 13.15 years
C) in 13.20 years
D) in 13.25 years

Question 20

If $2,000 is invested at 7 percent compounded monthly, what is the balance after 8 years?

A) $3,490.51
B) $3,495.65
C) $3,521.62
D) $3,601.84

Question 21

If $3,000 is invested at 10% compounded continuously, what is the balance after 9 years?

A) $7,378.79
B) $7,378.81
C) $7,391.90
D) $7,342.10

Question 22

If $1,000 is invested at 4 percent compounded continuously, what is the balance after 12 years?

A) $618.78
B) $1,040.00
C) $1,616.07
D) $1,601.03

Question 23

If $7,000 is invested at 6% interest compounded continuously, the balance after 10 years will be $12,725.31.

Question 24

If $7,000 is invested at 6% interest compounded monthly, the balance after 10 years will be $12,735.78.

Question 25

If $7,000 is invested at 7% interest for 20 years, there will be $400 more if the compounding is continuous rather than quarterly.

Question 26

How much money should be invested today at 5 percent compounded continuously so that 10 years from now it will be worth $10,000?

A) $6,065.31
B) $6,082.41
C) $6,083.43
D) $6,091.01

Question 27

How much money should be invested today at an annual interest rate of 7% compounded continuously so that 40 years from now it will be worth $5,000?

A) $ 4,661.97
B) $ 333.90
C) $ 82,223.23
D) $ 304.05

Question 28

How much money should be invested now at a yearly rate of 8 percent compounded quarterly so that 10 years from now the account will be worth $11,000?

A) $506.34
B) $5,095.13
C) $9,023.83
D) $4,981.79

Question 29

How much money, in dollars, should be invested today at 7 percent compounded continuously so that 5 years from now it will be worth $5,000?

Question 30

Sally needs $2,000 for a trip she plans to take 3 years from now. How much, in dollars, should she invest in an account that pays 7% interest compounded continuously in order to meet her goal?

Question 31

Some amount of money is to be invested today at 7% compounded continuously so that 10 years from now the account will be worth $5,000. The amount is $2,000.

Question 32

Find the present value, in dollars, of $10,000 payable 8 years from now if the annual interest rate is 7% compounded continuously.

Question 33

It is projected that t years from now, the population of a certain country will be $P ( t ) = 30 e ^ { 0.08 t }$ million. What will be the population in 40 years? Round your answer to two decimal places, if necessary.

A) 735.98 million
B) 330.7 million
C) 711.44 million
D) 760.51 million

Question 34

The effective interest rate (rounded to two decimal places) if the nominal rate is 7 percent compounded quarterly is 7.30%.

Question 35

Find the effective interest rate if the nominal rate is 6.5% compounded monthly. Round to one decimal place, if necessary.

A) 6.7%
B) 6.9%
C) 7.1%
D) 7.3%

Question 36

Find the effective interest rate if the nominal rate is 7 percent compounded continuously. Round to two decimal places, if necessary.

A) 7.05%
B) 7.25%
C) 7.45%
D) 7.55%

Question 37

The fraction of television sets manufactured by a certain company that are still in working condition after t years of use is approximately $f ( t ) = e ^ { - 0.1 t }$ . What fraction can be expected to fail before 5 years of use? Round your answer to two decimal places, if necessary.

A) 0.61
B) 1.65
C) 0.39
D) 1.00

Question 38

A statistical study indicates that the fraction of the can openers manufactured by a certain company that are still in working condition after t years of use is approximately $f ( t ) = e ^ { - 0.3 t }$ . The fraction that can be expected to fail during the 2nd year of use is about $\frac { 19 } { 100 }$ .

Question 39

The population density x miles from the center of a certain city is $D ( x ) = 9 e ^ { - 0.06 x }$ thousand people per square mile. What is the population density 5 miles from the center of this city? Round your answer to two decimal places.

A) 12.15 thousand people per square mile
B) 6.67 thousand people per square mile
C) 5.93 thousand people per square mile
D) 7.41 thousand people per square mile

Question 40

Determine the monthly car payment for a new car costing $19,723, if there is a down payment of $3,000 and the car is financed over a 10-year period at an annual rate of 8% compounded monthly.

A) $182.61
B) $239.29
C) $202.90
D) $217.31

Question 41

Evaluate the expression $e ^ { 2 \ln 8 - 3 \ln 6 }$ without using tables or calculators.

Question 42

Evaluation of $e ^ { 1 / 2 * \mathrm { n } 4.1 / 3 { *} \mathbf { n } 27 }$ without using tables or a calculator will show it is 6.

Question 43

Use logarithm rules to rewrite the expression in terms of log x and log y. $\log \left( x ^ { 1 / 3 } y ^ { 4 } \right)$

Question 44

Use logarithm properties to simplify the expression. $$\log _ { 7 } \left[ \frac { x ^ { 7 } ( 8 - 7 x ) ^ { 7 / 2 } } { \sqrt { x ^ { 2 } + x } } \right]$$

Question 45

Solve for x. Give an exact answer and one rounded to two decimal places. 9^x = 94

Question 46

Solve for x: $$\log _ { 64 } x = \frac { 1 } { 3 }$$

A) x = 4
B) $x = \frac { 64 } { 3 }$
C) $x = \frac { 1 } { 3 \log 64 }$
D) x = 8

Question 47

Solve for x. Round to three decimal places, if necessary. log₃ x = 4

A) x = 81
B) x = 64
C) x = 1.262
D) x = 0.792

Question 48

Solve for x. Round to three decimal places, if necessary. $\log _ { 2 } x = 3$

A) 8
B) 9
C) 1.26
D) 1.732

Question 49

Solve for x if log₃ x = 5.

Question 50

Solve for x if $\log _ { 6 } x = 3$ .

Question 51

Solve for x if $\log _ { 6 } ( 2 x ) = 5$ .

Question 52

If log₄ x - 2 = 5, then x = 3.161, when rounded to three decimal places.

Question 53

Solve for x if log₄ (2x) = 3.

Question 54

Solve for x: $\log _ { 2 } ( x - 1 ) = 5$

A) x = 32
B) x = 31
C) x = 33
D) x = 3,322

Question 55

Solve for x. Answer exactly and then round to three decimal places. ln(-3x + 38) = -1

Question 56

Solve for x: $\log _ { 6 } ( x - 1 ) = 5$ .

A) 7,777
B) 7,776
C) 7,775
D) 15,626

Question 57

If log₇ (2x) = 2 , then $x = \frac { 49 } { 2 }$ .

Question 58

If $\log _ { 5 } ( 7 x ) = 4$ , then $x = \frac { 62.5 } { 7 }$ .

Question 59

Solve for x. Give an exact answer and then round to two decimal places, if necessary. $7 ^ { - 8 x + 2 } = 85$

A) $- \frac { 1 } { 8 } \left( \frac { 85 } { 7 } - 2 \right) \approx - 1.27$
B) $- \frac { 1 } { 8 } \left( \frac { \ln 85 } { \ln 7 } - 2 \right) \approx - 0.04$
C) $- \frac { 1 } { 8 } ( \ln 85 - 2 \ln 7 ) \approx - 0.07$
D) No solution

Question 60

Solve for x: $a ^ { 2 x - 2 } = b$

A) $x = \frac { 1 } { 2 } \left( 2 + \frac { \ln b } { \ln a } \right)$
B) $\frac { \ln b } { 2 \ln a }$
C) $\left( \frac { \ln b } { \ln a } + 2 \right) ^ { 2 }$
D) $x = 1 + \frac { 1 } { 2 } \left( 2 + \frac { \ln b } { \ln a } \right)$

Question 61

Solve for x: $5 ^ { x } = e ^ { 3 }$

A) $x = \frac { 3 } { \ln 5 }$
B) x = ln 5
C) x = 3 ln 5
D) x = 3 - ln5

Question 62

Solve for x: $a ^ { 2 x - 1 } = b$

A) $x = \frac { \ln b } { 2 \ln a }$
B) $x = \frac { 1 } { 2 } \left( 1 + \frac { \ln b } { \ln a } \right)$
C) $x = \left( 1 + \frac { \ln b } { \ln a } \right) ^ { 2 }$
D) $x = 1 + \frac { \ln b } { 2 \ln a }$

Question 63

If $3 = e ^ { - 0.002 x }$ , then x is $\frac { \ln 3 } { 50 }$ .

Question 64

If $\frac { 1 } { 2 } A = 2 A e ^ { - 0.06 x }$ , then x is $\frac { \ln 4 } { 0.06 }$ .

Question 65

If $2 = e ^ { - 0.002 x }$ , then x is $\frac { \ln 2 } { 500 }$ .

Question 66

Solve for x: $5 = 7 - 3 e ^ { - 2 x }$

Question 67

Solve the given equation for x. $$0 = - 1 + 5 e ^ { - 7 x }$$

A) $- 7 e ^ { 5 }$
B) $- \frac { e ^ { 5 } } { 7 }$
C) $- \frac { \ln 5 } { 7 }$
D) $\frac { \ln 5 } { 7 }$

Question 68

Solve the given equation for x. $$- 5 = - 6 + 4 e ^ { 7 x }$$

A) $7 e ^ { 4 }$
B) $\frac { e ^ { 4 } } { 7 }$
C) $\frac { \ln 4 } { 7 }$
D) $- \frac { \ln 4 } { 7 }$

Question 69

Solve the given equation for x. $$7 = 6 + 4 e ^ { - 5 x }$$

A) $- 5 e ^ { 4 }$
B) $- \frac { e ^ { 4 } } { 5 }$
C) $- \frac { \ln 4 } { 5 }$
D) $\frac { \ln 4 } { 5 }$

Question 70

Solve the given equation for x. $$10 = 9 + 8 e ^ { 9 x }$$

A) $9 e ^ { 8 }$
B) $\frac { e ^ { 8 } } { 9 }$
C) $\frac { \ln 8 } { 9 }$
D) $- \frac { \ln 8 } { 9 }$

Question 71

Solve for x: $$6 = 8 - 3 e ^ { - 8 x }$$

Question 72

Solve for x: $$3 \ln x - \frac { 1 } { 4 } \ln x ^ { 3 } = 9$$

A) $x = e ^ { 4 }$
B) x = e
C) $x = e ^ { 9 }$
D) $x = e ^ { 3 }$

Question 73

Solve for x: $$2 \ln x - \frac { 1 } { 3 } \ln x ^ { 2 } = 4$$

A) x = e
B) $x = e ^ { 3 }$
C) $x = e ^ { 4 }$
D) $x = e ^ { 2 }$

Question 74

Solve for x to two decimal places: $$\ln \sqrt { x } = \frac { 1 } { 5 } ( \ln 8 + 2 \ln 3 )$$

Question 75

Solve for x to two decimal places: $$\ln \sqrt { x } = \frac { 1 } { 8 } ( \ln 7 + 2 \ln 4 )$$

Question 76

If ln x = 3(ln 2 - ln 7), then $x = \frac { 1 } { 6 }$ .

Question 77

Solve for x: $2 ^ { x } = e ^ { 3 }$

A) x = ln 2
B) x = 3 ln 2
C) x = 3 - ln 2
D) $x = \frac { 3 } { \ln 2 }$

Question 78

A bank compounds interest continuously. What nominal interest rate does it offer if $1,500 grows to $2,500 in 10 years? Round to two decimal places.

A) 5.00%
B) 5.07%
C) 5.11%
D) 5.13%

Question 79

It takes 8 years for money to double with quarterly compounding if the annual interest rate is 8%.

Question 80

How quickly will money triple if it is invested at 7% interest compounded continuously? Round your answer to two decimal places.

A) 15.69 years
B) 15.71 years
C) 15.73 years
D) 15.75 years