Deck 4: Exponential and Logarithmic Functions

Question 1

Graph the function , not by plotting points but by applying your knowledge of the general shape of graphs of the form . State the domain, range, and asymptote of the function.

Answer 1

Domain , Range , Asymptote

Question 2

If $4500 is invested at an interest rate of 6.25% per year, compounded continuously, find the value of the investment after the given number of years.
(a) 3 years
(b) 6 years
(c) 9 years

Answer 2

a) $5428.04 (b) $6547.46 (c) $7897.75

Question 3

Graph the function, not by plotting points, but by starting from the graph of . State the domain, range, and asymptote.

Answer 3

Domain: ; Range: ; Asymptote:

Question 4

A radioactive substance decays in such a way that the amount of mass remaining after t days is given by the function where is measured in kilograms. Round your answers to three decimal places.
a) Find the mass at time .
b) How much of the mass remains after 100 days?

Question 5

Sketch the graph of the function by making a table of values. Use a calculator if necessary.

Question 6

If , show that .

Question 7

Graph the function, not by plotting points, but by starting from the graph of . State the domain, range, and asymptote.

Question 8

If $2500 is invested at an interest rate of 6.25% per year, compounded continuously, find the value of the investment after the given number of years.
(a) 3 years
(b) 6 years
(c) 9 years

Question 9

Graph the function , not by plotting points but by applying your knowledge of the general shape of graphs of the form . State the domain, range, and asymptote of the function.

Question 10

Sketch the graph of the function by making a table of values. Use a calculator if necessary.

Question 11

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.

Question 12

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.

Question 13

Graph the function , not by plotting points but by applying your knowledge of the general shape of graphs of the form . State the domain, range, and asymptote of the function.

Question 14

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.

Question 15

Find the exponential function whose graph is given.

Question 16

A radioactive substance decays in such a way that the amount of mass remaining after t days is given by the function where is measured in kilograms. Round your answers to three decimal places.
a) Find the mass at time .
b) How much of the mass remains after 50 days?

Question 17

For the function , ( ), use a graphing calculator to find, correct to two decimal places,
(a) the intervals on which the function is increasing or decreasing, and
(b) the range of the function.

Question 18

If $10,000 is invested at an interest rate of 5% per year, compounded semiannually, find the value of the investment after the given number of years.
(a) 5 years
(b) 10 years
(c) 15 years

Question 19

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.

Question 20

If $10,000 is invested at an interest rate of 4% per year, compounded semiannually, find the value of the investment after the given number of years.
(a) 5 years
(b) 10 years
(c) 15 years

Question 21

Use the definition of the logarithmic function to find if .

Question 22

If $12,000 is invested in an account for which interest is compounded continuously, find the amount of the investment at the end of 12 years for the following interest rates.
(a) 2%
(b) 7%

Question 23

Evaluate the expression. $\log _ { \sqrt { 7} } 49$

A) $4$
B) $7$
C) $\frac { 1 } { 2 }$
D) $- \frac { 1 } { 2 }$
E) $\frac { 1 } { 4 }$

Question 24

If $10,000 is invested in an account at 5% per year, find the amount of the investment at the end of 12 years for the following compounding methods.
(a) Continuously
(b) Daily

Question 25

Evaluate the expression. $\log _ { 7 } 1$

A) $- 1$
B) $1$
C) $3$
D) $5$
E) $0$

Question 26

Use a calculator to evaluate the expression $\ln ( 4 - \sqrt { 10 } )$ correct to four decimal places.

A) $0.1770$
B) $0.7690$
C) $- 0.1771$
D) $- 0.7699$
E) $- 1.776$  

Question 27

Express the equation $\log _ { 13 } 1 = 0$ in exponential form.

A) $13 ^ { 1} = 0$
B) $0 ^ { 13 } = 1$
C) $13 ^ { \mathrm { D } } = 1$
D) $1 ^ { 13 } = 1$
E) $13 ^ { 1 } = 13$

Question 28

Express the equation $6 ^ { 3 } = 216$ in logarithmic form.

A) $\log _ { 2 } 216 = 3$
B) $\log _ { 3 } 216 = 6$
C) $\log _ { 6 } 216 = 3$
D) $\log _ { 6 } 216 = 36$
E) $\log _ { 6 } 36 = 2$

Question 29

Evaluate the expression. $\ln \left( e ^ { n + 1 } \right)$

A) $- 1$
B) $\pi + 1$
C) $e ^ { n }$
D) $\pi$
E) $\ln ( x + 1 )$

Question 30

Use the definition of the logarithmic function to find if .

Question 31

The air resistance of a free falling object is proportional to its velocity. Use the equation of downward velocity given by , where is time and is measured in ft/s, to find the following.
(a) The initial velocity of the object.
(b) The velocity of the object after s and
s, if
.
(c) The object's maximum or terminal velocity, and a graph of the object's velocity.

Question 32

Evaluate the expression. $\log _ { 4 } 256$

A) $3$
B) $4$
C) $5$
D) $125$
E) 625 

Question 33

If $6000 is invested in an account for which interest is compounded continuously, find the amount of the investment at the end of 12 years for the following interest rates.
(a) 2%
(b) 7%

Question 34

Use a calculator to evaluate the expression $\log 0.5 \text {, }$ to four decimal places.

A) $- 0.2241$
B) $- 0.1042$
C) $- 0.0314$
D) $- 0.9421$
E) $- 0.3010$

Question 35

Use a calculator to evaluate the expression $\log 13$ to four decimal places.

A) $1.2465$
B) $1.5493$
C) $1.6489$
D) $1.9954$
E) $1.1139$

Question 36

The height from ground level to a hanging cable suspended between two utility poles can be described by the function , where is the horizontal distance in feet, measured from the origin, and is the lowest point of the hanging cable.
(a) Find the height of a cable, correct to two decimal places, feet to the right of its origin and its lowest point
feet above ground level.
(b) Find the height of a cable, correct to two decimal places, feet to the left of its origin and its lowest point
feet above ground level.

Question 37

Evaluate the expression.

A) $2$
B) $\ln ( 2 )$
C) $\ln \left( \frac { 1 } { 2 } \right)$
D) $2 e$
E) $e ^ { 2 }$  

Question 38

Use the definition of the logarithmic function to find if .

Question 39

Express the equation $\log _ { 32 } 8 = \frac { 3 } { 5 }$ in exponential form.

A) $3 ^ { 5 } = 8$
B) $\frac { 2 ^ { 3 } } { 2 } = 16$
C) $8 = 3 ^ { 2 }$
D) $32 ^ { 3 / 5 } = 8$
E) $8 ^ { 5 / 3 } = 32$  

Question 40

Use a calculator to evaluate the expression $\log _ { 10 } \left( \frac { 4 } { 5 } \right)$ correct to four decimal places.

A) $0.0970$
B) $- 0.2231$
C) $0.1204$
D) $- 0.0969$
E) $- 0.1204$

Question 41

The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If is the original amount of carbon-14 and D is the amount remaining, then the artifact's age A (in years) is given by Find the age of an object if the amount D of carbon-14 that remains in the object is 85% of the original amount .

Question 42

Use the Laws of Logarithms to rewrite the expression in a form with no logarithm of a product, quotient, root, or powers.

Question 43

Use the Laws of Logarithms to rewrite the expression in a form with no logarithm of a product, quotient, root, or powers.  

Question 44

Use the Laws of Logarithms to rewrite $\log _ { 11 } \sqrt [ 3 ] { 10 }$ in a form with no logarithms of products, quotients, or powers.

A) $\frac { 1 } { 11 } \log _ { 3 } 10$
B) $\log _ { 11 } 10 - \log _ { 11 } 3$
C) $\sqrt [ 3 ] { \log _ { 11 } 10 }$
D) $\frac { 1 } { 3 } \log _ { 11 } 10$
E) $\frac { 1 } { 3 } \log _ { 1 } 3$  

Question 45

Evaluate the expression. $\log _ { 12 } 3 + \log _ { 12 } 48$

A) $1$
B) $2$
C) $3$
D) $4$
E) $5$  

Question 46

A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time t (in hours) required for the colony to grow to N bacteria is given by Find the time required for the colony to grow to five million bacteria.

Question 47

Evaluate the expression. $\log _ {5 } 625 - \log _ { 5 } 125$

A) $1$
B) $2$
C) $3$
D) $- 3$
E) $- 1$  

Question 48

A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time t (in hours) required for the colony to grow to N bacteria is given by Find the time required for the colony to grow to two million bacteria.

Question 49

Use the Laws of Logarithms to rewrite the expression in a form with no logarithm of a product, quotient, root, or powers.  

Question 50

Evaluate the expression. $\log \sqrt { 0.0001 }$

A) $- 1$
B) $2$
C) $- 2$
D) $- \frac { 1 } { 2 }$
E) $\frac { 1 } { 3 }$  

Question 51

Evaluate the expression.  

Question 52

Evaluate the expression. $\log _ { 2 } 192 - \log _ { 2 } 3$

A) $3$
B) $4$
C) $5$
D) $6$
E) $7$

Question 53

Graph the function , not by plotting points but by applying your knowledge of the shape of the log function. State domain, range, and asymptote.  

Question 54

The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If is the original amount of carbon-14 and D is the amount remaining, then the artifact's age A (in years) is given by Find the age of an object if the amount D of carbon-14 that remains in the object is 50% of the original amount .

Question 55

The time required to double the amount of an investment at an interest rate r compounded continuously is given by Find the time required to double an investment at 2%, 3%, and 4%.

Question 56

(a)Find the inverse function. (b) What is the domain of the inverse function?

Question 57

Use the Laws of Logarithms to rewrite the expression in a form with no logarithm of a product, quotient, root, or powers.  

Question 58

The time required to double the amount of an investment at an interest rate r compounded continuously is given by Find the time required to double an investment at 4%, 5%, and 6%.

Question 59

Graph the function not by plotting points but by applying your knowledge of the shape of the log function. State domain, range, and asymptote.

Question 60

Use the Laws of Logarithms to rewrite $\ln ( e / x )$ in a form with no logarithms of products, quotients, or powers.

A) $2 + 2 \ln x$
B) $1 - \ln x$
C) $e \ln x$
D) $- \ln x$
E) $1 + \ln x$

Question 61

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms.

Question 62

For a rat learning to run a maze, the learning curve is given by , where is the time it takes to run the maze in seconds after days. Use a graphing calculator or computer to graph of the learning curve and determine after how many days the rat is able to run the maze in seconds.

Question 63

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms.

Question 64

Assume that the rat population in a certain urban area behaves according to the logistic growth function where is the number of rats in the hundreds, and is measured in months.(a) Find the initial rat population.(b) Find the rat population after month.(c) After how many months will the rat population triple from its initial size?

Question 65

Rudy wants to invest $\$ 1000$ in savings certificates that bear an interest rate of $7.5 \%$ compounded semiannually. How long will it take for the amount to be $\$ 1500 ?$

A) $2.54$ yrs
B) $3.08$ yrs
C) $5.51$ yrs
D) $9.25$ yrs
E) $10.36$ yrs 

Question 66

Find the solution of the equation $$\left( \frac { 3 } { 5 } \right) ^ { 7 x } = 12$$ correct to four decimal places.

A) $- 2.4322$
B) $- 3.0501$
C) $- 2.7593$
D) $- 2.1175$
E) $- 1.7659$  

Question 67

Find the solution of the equation $3 ^ { x + 2 } = 5 ^ { 2 x }$ correct to four decimal places.

A) $- 0.0365$
B) $0.0265$
C) $0.3785$
D) $0.3932$
E) $1.0363$  

Question 68

A 16-g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by , where is measured in grams. After how many days is there only 4 g remaining?

Question 69

Solve the logarithmic equation for .

Question 70

Use the change of base formula and a calculator to evaluate correct to six decimal places.

Question 71

Use a graphing calculator to find all solutions of the equation correct to two decimal places.

Question 72

Find the solution of the equation $2 ^ { x/2 } = 0.01$ correct to four decimal places.

A) $- 53.1508$
B) $- 29.0743$
C) $- 25.1508$
D) $- 19.0828$
E) $- 7.0523$  

Question 73

Solve the equation .

Question 74

Show that . 

Question 75

Use the change of base formula and a calculator to evaluate correct to six decimal places.

Question 76

Solve the equation .

Question 77

A sum of $1250 was invested for 8 years, and the interest compounded quarterly. If this sum amounted to in the given time, what was the interest rate?

Question 78

How long will it take for an investment of to double in value if the interest rate is per year compounded continuously?

Question 79

A 12-g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by , where is measured in grams. After how many days is there only 4 g remaining?

Question 80

Find the solution of the exponential equation , correct to four decimal places.