Deck 20: Exponential and Logarithmic Functions

Question 1

Graph the exponential function over the given domain of $x$ .
$y=4-4 e^{-x}(x=0$ to 4 $)$

Answer 1

Question 2

Graph the exponential function over the given domain of $x$ .
$y=e^{x / 2}-e^{-x / 2} \quad(x=-4$ to +4 $)$

Answer 2

Question 3

A student invests $\$ 4,000$ in an account that pays $5 \%$ interest per year, compounded annually. How much interest does the student earn if he leaves the money in the account for three years?

Answer 3

$\$ 630.50$

Question 4

A person has $\$ 25,000$ is invested in an account that pays $6 \%$ interest per year, compounded monthly. What is the value of the account after ten years?

Question 5

What amount needs to be invested now so that after 5 years the account has a balance of $\$ 40,000$ ? The account has a $3 \%$ interest rate per year, compounded annually.

Question 6

How much money does a company have to invest in an account that pays $6 \%$ per year, compounded annually, to have $\$ 12,000$ after three years?

Question 7

Find the amount to which $\$ 100$ will accumulate in 5 years at a compound interest rate of $5 \%$ per year, compounded annually.

Question 8

Find the amount to which $\$ 250$ will accumulate in 7 years at a compound interest rate of $3 \%$ per year, compounded annually.

Question 9

If an investment will double its value in 4 years, what annual compound interest rate (compounded annually) is being earned?

Question 10

If an investment grows $75 \%$ in 4 years, what annual compound interest rate (compounded annually) is being earned?

Question 11

A father invests $\$ 7500$ in a savings account that earns $5.4 \%$ interest compounded annually when his child is born. What will the value of the account be after 18 years when the child is ready to go to college or university?

Question 12

If an amount of money $P$ is deposited annually for $t$ years into an account with an interest rate $r$ ,it will accumulate to a value $V$ according to the equation $V=P\left(\frac{(1+r)^{t}-1}{r}\right)$ . If someone pays $\$ 5000$ annually, what will the value of the account be after 20 years?

Question 13

The value of a mortgage $M$ can be found using $M=P\left(\frac{1-\left(1+\frac{r}{12}\right)^{-12 t}}{\frac{r}{12}}\right)$ , where $P$ is the monthly payment, $r$ is the interest rate compounded monthly and $t$ is the number of years for the mortgage. What would the monthly payment be for a $\$ 250,000$ mortgage with a $6 \%$ interest compounded monthly for 20 years?

Question 14

A bacteria grows exponentially at a rate of $25.0 \%$ per hour for 5 hours. Find the final amount if the initial amount is 1000 .

Question 15

A culture of bacteria has a population that doubles every hour. Find the population of the bacteria after 4 hours if the initial population is 750 .

Question 16

What is the expected cost of a laptop computer in 5 years? The current cost of a laptop computer is $\$ 800$ and the rate of inflation is $8 \%$ per year.

Question 17

A room initially $20.0^{\circ} \mathrm{C}$ above the outside temperature cools exponentially at a rate of $20 \%$ per hour. Find the temperature of the room (above the outside temperature) after $150 \mathrm{~min}$ .

Question 18

A quantity grows exponentially from 0 to a maximum of 100 at a rate of $8 \%$ per second. Write the equation for the quantity at any second. What is the quantity at $5.0 \mathrm{~s}$ ?

Question 19

A certain company comes out with a new product and the number of investors in the company grows exponentially at a rate of $10 \%$ per week. After three weeks, there are 1,290 investors in the company. How many investors were there before the three weeks?

Question 20

In a certain chemical reaction the concentration of a certain product molecule $C$ is given by $C=2\left(1-e^{-0.13 t}\right)$ moles/liter, where $t$ is the time the reaction has been running in seconds. Find $C$ after the reaction has run for eleven seconds.

Question 21

A certain isotope of silver, ${ }^{110} \mathrm{Ag}$ decays at a rate of $2.8 \%$ per second. After 50.0 seconds, what percent of the original isotope remains?

Question 22

The half-life of phosphorus-32 is 14.29 days and decays according to the formula $F=A e^{-0.0485 t}$ , where $F$ is the final amount, $A$ is the initial amount and $t$ is the time in days. How much phosphorus- 32 is left after 60 days, if the initial amount was $25.0 \mathrm{mg}$ ?

Question 23

The voltage across a particular capacitor grows exponentially from $0 \mathrm{~V}$ to a maximum of $40.0 \mathrm{~V}$ , at a rate of $8 \%$ per second. What is the voltage at 5.00 seconds?

Question 24

Convert to logarithmic form: $6^{5}=7,776$

Question 25

Convert to logarithmic form: $3^{5}=243$

Question 26

Convert to logarithmic form: $x^{4}=261$

Question 27

Convert to logarithmic form: $h^{2 / 3}=8.8$

Question 28

Convert to logarithmic form: $3.1^{P}=2.4$

Question 29

Convert to logarithmic form: $3^{X}=9$

Question 30

Convert to logarithmic form: $e^{2 x}=y$

Question 31

Convert to logarithmic form: $10^{x}=5 y$

Question 32

Convert to exponential form: $\log _{3} 81=4$

Question 33

Convert to exponential form: $\log _{0.5} 16=-4$

Question 34

Convert to exponential form: $\log _{3} 24=y$

Question 35

Convert to exponential form: $\log _{X} 5=z$

Question 36

Convert to exponential form: $\log _{3} x=258$

Question 37

Convert to exponential form: $\log _{a} c=b$

Question 38

Convert to exponential form: $\log _{x+2} 35=100$

Question 39

Convert to exponential form: $\log _{2.5}(x-3)=-5.5$

Question 40

Convert to exponential form: $\log _{5} x=3 y$

Question 41

Find the common logarithm of each number to four decimal places:
(a) 28.1
(b) 9.51
(c) 10.003
(d) 0.0081

Question 42

Find the common logarithm of each number to four decimal places:
(a) 0.5
(b) 12.75
(c) 291
(d) 0.0132

Question 43

Find $x$ to three significant digits if the common logarithm of $x$ is equal to:
(a) -2.087
(b) 5.917
(c) -1.015
(d) 3.617

Question 44

Find $x$ to three significant digits if the common logarithm of $x$ is equal to:
(a) 2.4
(b) -0.83
(c) -12.4
(d) 1.32

Question 45

Find the natural logarithm of each number to four decimal places:
(a) 53.1
(b) 8080
(c) 0.00381
(d) 2.01

Question 46

Find the natural logarithm of each number to four decimal places:
(a) 27
(b) 6.55
(c) 0.14
(d) 672

Question 47

Find $x$ to four significant digits if the natural logarithm of $x$ is equal to:
(a) 2.1919
(b) -2.1919
(c) 4.010
(d) -0.397

Question 48

Find $x$ to three significant digits if the natural logarithm of $x$ is equal to:
(a) 1.5
(b) 0.091
(c) -2.37
(d) -0.64

Question 49

Graph the following function from $x=1 / 4$ to $x=4$ : $y=-0.3 \log _{4} 2 x$

Question 50

Graph the following function from $x=1 / 4$ to $x=4$ : $y=-2.5 \log _{3}(4-x)$

Question 51

Convert to logarithmic form: $4^{x / 2}=25$

Question 52

Convert to logarithmic form: $e^{\pi t}=v$

Question 53

Convert to exponential form: $W=\log T$

Question 54

Convert to exponential form: $y=\ln 2 x$

Question 55

Write as the sum or difference of two or more logarithms: $\log \frac{9}{8}$

Question 56

Write as the sum or difference of two or more logarithms: $\log \frac{9 b}{3}$

Question 57

Write as the sum or difference of two or more logarithms:
$\log \frac{2 a b}{c}$

Question 58

Write as the sum or difference of two or more logarithms:
$\log \frac{2 x^{2} y}{3 z^{3}}$

Question 59

Write as the sum or difference of two or more logarithms:
$\log \frac{2 \sqrt{x}}{y^{3}}$

Question 60

Write as the sum or difference of two or more logarithms: $\log \frac{x^{4}}{y \sqrt[3]{z}}$

Question 61

Express as a single logarithm with a coefficient of $1: 4 \log z+5 \log y-\log w$

Question 62

Express as a single logarithm with a coefficient of 1: $2 \log 3$

Question 63

Express as a single logarithm with a coefficient of 1 :
$3 \log \frac{a}{p}+\log \frac{b}{q}-\log \frac{c}{r}$

Question 64

Express as a single logarithm with a coefficient of 1 :
$\frac{\log z}{2}+5 \log x-2 \log y$

Question 65

Express as a single logarithm with a coefficient of $1: 4 \log x-3 \log y-2 \log z$

Question 66

Rewrite the equation so that it contains no logarithms: $2 \log x-3 \log y=1$

Question 67

Rewrite the following equation so that it contains no logarithms: $\log (x+y)+\log 2=\log \left(x^{2}-y^{2}\right)$

Question 68

Rewrite the equation so that it contains no logarithms: $2 \log _{3} x-\log _{3}\left(x^{2}-1\right)=1$

Question 69

Simplify: $10^{2 \log x}$

Question 70

Simplify: $e^{-3 \ln x}$

Question 71

Simplify: $\log _{5} 5^{y}$

Question 72

Rewrite the equation so that it contains no logarithms: $2 \ln x=2+\ln (x+1)$

Question 73

Rewrite the equation so that it contains no logarithms: $3 \log x+\log (x-2)=1+\log (x+3)$

Question 74

Solve for $x$ to three significant digits: $3^{X}=8$

Question 75

Solve for $x$ to three significant digits: $10^{2 x}=3^{x-3}$

Question 76

Solve for $x$ to three significant digits: $(9.90)^{x-3}=11.0$

Question 77

Solve for $x$ to three significant digits: $4^{-2 x}=2(4)^{-3 x}$

Question 78

Solve for $x$ to three significant digits: $4(10)^{x-1}-3(10)^{x-2}=1$

Question 79

Solve for $x$ to three significant digits: $e^{4 x}-4 e^{2 x}+3=0$

Question 80

Solve for $x$ to three significant digits: $\left(\frac{2}{3}\right)^{x+1}\left(\frac{3}{2}\right)=\frac{2}{3}$