Deck 4: Exponential and Logarithmic Functions

Question 1

Solve for x: $a ^ { 7 x - 1 } = b$

A) $\left( \frac { \ln b } { \ln a } + 1 \right) ^ { 7 }$
B) $x = \frac { 1 } { 7 } \left( 1 + \frac { \ln b } { \ln a } \right)$
C) $x = 1 + \frac { 1 } { 7 } \left( 1 + \frac { \ln b } { \ln a } \right)$
D) $\frac { \ln b } { 7 \ln a }$

Answer 1

$x = \frac { 1 } { 7 } \left( 1 + \frac { \ln b } { \ln a } \right)$

Question 2

Solve for x.Round to three decimal places,if necessary. $\log _ { 4 } x = 5$

A)1)495
B)1)32
C)1,024
D)625

Answer 2

1,024

Question 3

An efficiency expert hired by a manufacturing firm has compiled the following data relating workers' output to their experience: Experience (months)
$$\begin{array} { l l l } \text { Experience (months) } & 0 & 2 \\\text { Output (units per hour) } & 300 & 510\end{array}$$
The expert believes that the output Q is related to experience t by a function of the form $Q ( t ) = 600 - A e ^ { - k t }$ .Find the function of this form that fits the data.Round numbers to four decimal places,if necessary.

A) $Q ( t ) = 600 - 510 e ^ { 0.6020 t }$
B) $Q ( t ) = 600 - 510 e ^ { - 0.6020 t }$
C) $Q ( t ) = 600 - 300 e ^ { - 0.6020 t }$
D) $Q ( t ) = 600 - 300 e ^ { 0.6020 t }$

Answer 3

$Q ( t ) = 600 - 300 e ^ { - 0.6020 t }$

Question 4

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

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

Question 5

Evaluate the given expression. $\left( \frac { 256 } { 625 } \right) ^ { 5 / 4 }$

A) $\frac { 1,024 } { 3,125 }$
B) $\frac { 4 } { 5 }$
C) $\frac { 1,024 } { 625 }$
D) $\frac { 256 } { 625 }$

Question 6

Find the derivative of $\ln \left[ \left( \ln x ^ { 6 } \right) ^ { 7 } \right]$ .

A) $\frac { 42 } { x \ln x }$
B) $\frac { 42 } { \ln ( \ln x ) }$
C) $\frac { 7 } { x \ln x }$
D) $\frac { 42 } { x + \ln x }$

Question 7

Find all real numbers x that satisfy the given equation. 3^x2^3x = 13,824

A)15
B)12
C)9
D)3

Question 8

The consumer demand for a certain commodity is $D ( p ) = 8,000.00 e ^ { - 057 p }$ units per month when the market price is p dollars per unit.Express consumers' total monthly expenditure for the commodity as a function of p and determine the market price that will result in the greatest consumer expenditure.

A)$8,000.00
B)$14.04
C)$26.32
D)$1.75

Question 9

It is projected that t years from now,the population of a certain country will be $P ( t ) = 70 e ^ { 0.01 t }$ million.What will be the population in 20 years? Round your answer to two decimal places,if necessary.

A)140.96 million
B)86.72 million
C)85.5 million
D)84.28 million

Question 10

Solve for x: $4 \ln x - \frac { 1 } { 5 } \ln x ^ { 4 } = 16$

A)x = e
B) $x = e ^ { 16 }$
C) $x = e ^ { 4 }$
D) $x = e ^ { 5 }$

Question 11

Use logarithmic differentiation to find $f ^ { \prime } ( x )$ . $f ( x ) = \sqrt [ 8 ] { \frac { 6 x + 5 } { 8 + 6 x } }$

A) $f ^ { \prime } ( x ) = \frac { 1 } { 8 } \left( \frac { 6 } { 6 x + 5 } - \frac { 6 } { 8 + 6 x } \right) f ( x )$
B) $f ^ { \prime } ( x ) = f ( x ) \left( \frac { 6 x + 5 } { 8 + 6 x } \right) ^ { - 7 / 8 }$
C) $f ^ { \prime } ( x ) = \left( \frac { 6 } { 6 x + 5 } - \frac { 6 } { 8 + 6 x } \right) f ( x )$
D) $f ^ { \prime } ( x ) = \left( \frac { 6 x + 5 } { 8 + 6 x } \right) ^ { - 7 / 8 }$

Question 12

Find all real numbers x that satisfy the given equation. $\left( \frac { 1 } { 81 } \right) ^ { x - 1 } = 3 ^ { 4 - 3 x ^ { 2 } }$

A)0, $- \frac { 4 } { 3 }$
B)0, $\frac { 4 } { 3 }$
C) $- \frac { 4 } { 3 }$
D)0

Question 13

Differentiate the given function. $f ( x ) = e ^ { - 6 x }$

A) $x = - 6 x e ^ { - 6 x }$
B) $x = e ^ { - 6 x }$
C) $x = - 6 e ^ { - 6 x - 1 }$
D) $x = - 6 e ^ { - 6 x }$

Question 14

Let $f ( t ) = \frac { 19 } { 3 + 4 e ^ { - t / 10 } }$ .Which value of t corresponds to a possible inflection point for f (t)?

A)There is no inflection point.
B) $\frac { 3 } { 4 } \ln ( 110 )$
C) $\ln \left( \frac { 3 } { 4 } \right)$
D) $110 \ln \left( \frac { 4 } { 3 } \right)$

Question 15

Determine the monthly car payment for a new car costing $18,228,if there is a down payment of $6,000 and the car is financed over a 6-year period at an annual rate of 9% compounded monthly.

A)$328.57
B)$220.42
C)$231.50
D)$198.37

Question 16

If $1,500 is invested at 9 percent compounded continuously,what is the balance after 13 years?

A)$4,598.71
B)$465.55
C)$1,635.00
D)$4,832.99

Question 17

Differentiate the given function. $f ( x ) = \ln x ^ { 3 }$

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

Question 18

A radioactive substance decays exponentially.If 700 grams were present initially and 300 grams are present 100 years later,how many grams will be present after 400 years? Round your answer to two decimal places,if necessary.

A)22.37 grams
B)21.12 grams
C)0)00 grams
D)23.62 grams

Question 19

Differentiate the given function. $f ( x ) = 35 - 3 e ^ { - 0.06 x }$

A) $3 x e ^ { - 0.06 x }$
B) $- 0.18 e ^ { - 0.06 x }$
C) $0.18 e ^ { - 006 x }$
D) $- 3 x e ^ { - 0.06 x }$

Question 20

Solve for x: $\log _ { 5 } ( x - 1 ) = 3$ .

A)125
B)244
C)126
D)124

Question 21

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) $6 \left( 3 ^ { 5 } - 15 \ln 3 \right)$

Question 22

Consider the function $f ( x ) = e ^ { - ( x - 6 ) ^ { 2 } / 5 }$ .For what value of x does this function attain its maximum value,and what is the maximum function value? Round maximum function value to two decimal places,if necessary.

A)x = 6,f (6)= 1
B) $x = \frac { 3 } { 5 }$ , $f \left( \frac { 3 } { 5 } \right)$ $\approx$ 0.00
C)There is no maximum.
D) $x = \frac { 6 } { 5 }$ , $f \left( \frac { 6 } { 5 } \right)$ $\approx$ 0.01

Question 23

Suppose your family owns a rare book whose value t years from now will be $V ( t ) = 9 e ^ { \sqrt { 0.8 t } }$ dollars.If the prevailing interest rate remains constant at 6% per year compounded continuously,when will it be most advantageous for your family to sell the book and invest the proceeds? Round your answer to two decimal places.

A)125.00 years
B)48.61 years
C)194.44 years
D)55.56 years

Question 24

A traffic accident was witnessed by $\frac { 1 } { 14 }$ of the residents of a small town.The number of residents who had heard about the accident t hours later is given by a function of the form $\frac { B } { 1 + C e ^ { - k t } }$ ,where B is the population of the town.If $\frac { 1 } { 4 }$ of the residents had heard about the accident after 1 hours,how long did it take for $\frac { 1 } { 2 }$ of the residents to hear the news? Round your answer to two decimal places.

A)0.87 hours
B)0.44 hour
C)1.75 hours
D)3.50 hours