Precalculus

Jay Abramson

Chapter 4

Exponential and Logarithmic Functions - all with Video Answers

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Section 1

Exponential Functions

Problem 1

Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function.

Yousef Sheikh

Yousef Sheikh

Numerade Educator

Problem 2

Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain.

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 3

The Oxford Dictionary defines the word nominal as a value that is "stated or expressed but not necessarily corresponding exactly to the real value." Develop a reasonable argument for why the term nominal rate is used to describe the annual percentage rate of an investment account that compounds interest.

Yousef Sheikh

Numerade Educator

Problem 4

For the following exercises, identify whether the statement represents an exponential function. Explain.
The average annual population increase of a pack of wolves is $25 .$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 5

For the following exercises, identify whether the statement represents an exponential function. Explain.
A population of bacteria decreases by a factor of $\frac{1}{8}$ every 24 hours.

Yousef Sheikh

Numerade Educator

Problem 6

For the following exercises, identify whether the statement represents an exponential function. Explain.
The value of a coin collection has increased by 3.25$\%$ annually over the last 20 years.

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 7

For the following exercises, identify whether the statement represents an exponential function. Explain.
For each training session, a personal trainer charges his clients $\$ 5$ less than the previous training session.

Yousef Sheikh

Numerade Educator

Problem 8

For the following exercises, identify whether the statement represents an exponential function. Explain.
The height of a projectile at time $t$ is represented by the function $h(t)=-4.9 t^{2}+18 t+40$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 9

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.)
Which forest's population is growing at a faster rate?

Yousef Sheikh

Numerade Educator

Problem 10

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.)
Which forest had a greater number of trees initially? By how many?

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 11

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.)
Assuming the population growth models continue to represent the growth of the forest will have a greater number of trees after 20 years? By how many?

Yousef Sheikh

Numerade Educator

Problem 12

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.)
Assuming the population growth models continue to represent the forest will have a greater number of trees after 100 years? By how many?

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 13

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.)
Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the long-term validity of the exponential growth model?

Yousef Sheikh

Numerade Educator

Problem 14

For the following exercises, determine whether the equation repponential grownth, exponential decay, or neither. Explain.
$$
y=300(1-t)^{5}
$$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 15

For the following exercises, determine whether the equation repponential grownth, exponential decay, or neither. Explain.
$$
y=220(1.06)^{x}
$$

Yousef Sheikh

Numerade Educator

Problem 16

For the following exercises, determine whether the equation repponential grownth, exponential decay, or neither. Explain.
$$
y=16.5(1.025)^{\frac{1}{x}}
$$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 17

For the following exercises, determine whether the equation repponential grownth, exponential decay, or neither. Explain.
$$
y=11,701(0.97)^{t}
$$

Yousef Sheikh

Numerade Educator

Problem 18

For the following exercises, find the formula for an exponential function that passes through the two points given.
$$
(0,6) \text { and }(3,750)
$$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 19

For the following exercises, find the formula for an exponential function that passes through the two points given.
$$
(0,2000) \text { and }(2,20)
$$

Yousef Sheikh

Numerade Educator

Problem 20

For the following exercises, find the formula for an exponential function that passes through the two points given.
$$
\left(-1, \frac{3}{2}\right) \text { and }(3,24)
$$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 21

For the following exercises, find the formula for an exponential function that passes through the two points given.
$$
(-2,6) \text { and }(3,1)
$$

Marcella Sippey

Marcella Sippey

Numerade Educator

Problem 22

For the following exercises, find the formula for an exponential function that passes through the two points given.
$$
(3,1) \text { and }(5,4)
$$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 23

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

Oluwadamilola Ameobi

Oluwadamilola Ameobi

Numerade Educator

Problem 24

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 25

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

Oluwadamilola Ameobi

Oluwadamilola Ameobi

Numerade Educator

Problem 26

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 27

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

Oluwadamilola Ameobi

Oluwadamilola Ameobi

Numerade Educator

Problem 28

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
After a certain number of years, the value of an investment account is represented by the equation $10,250\left(1+\frac{0.04}{12}\right)^{120} .$ What is the value of the account?

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 29

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
What was the initial deposit made to the account in the previous exercise?

Yousef Sheikh

Numerade Educator

Problem 30

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
How many years had the account from the previous exercise been accumulating interest?

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 31

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
An account is opened with an initial deposit of $\$ 6,500$ and earns 3.6$\%$ interest compounded semi-annually. What will the account be worth in 20 years?

Yousef Sheikh

Numerade Educator

Problem 32

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
How much more would the account in the previous exercise have been worth if the interest were compounding weekly?

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 33

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
Solve the compound interest formula for the principal, $P$

Yousef Sheikh

Numerade Educator

Problem 34

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
Use the formula found in the previous exercise the initial deposit of an account that is worth $\$ 14,472.74$ after earning 5.5$\%$ interest compounded monthly for 5 years. (Round to the nearest dollar.)

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 35

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
How much more would the account in the previous two exercises be worth if it were earning interest for 5 more years?

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 36

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
Use properties of rational exponents to solve the compound interest formula for the interest rate, $r$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 37

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semiannually, had an initial deposit of $\$ 9,000$ and was worth $\$ 13,373.53$ after 10 years.

Yousef Sheikh

Numerade Educator

Problem 38

For the following exercises, use the compound interest formula, $A(t)=P\left(1+\frac{r}{n}\right)^{n t}$
Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of $\$ 5,500$ , and was worth $\$ 38,455$ after 30 years.

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 39

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.
$$
y=3742(e)^{0.75 t}
$$

Yousef Sheikh

Numerade Educator

Problem 40

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.
$$
y=150(e)^{\frac{3.25}{t}}
$$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 41

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.
$$
y=2.25(e)^{-2 t}
$$

Yousef Sheikh

Numerade Educator

Problem 42

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.
Suppose an investment account is opened with an initial deposit of $\$ 12,000$ earning 7.2$\%$ interest compounded continuously. How much will the account be worth after 30 years?

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 43

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.
How much less would the account from Exercise 42 be worth after 30 years if it were compounded monthly instead?

Joanna Quigley

Numerade Educator

Problem 44

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
$$
f(x)=2(5)^{x}, \text { for } f(-3)
$$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 45

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
$$
f(x)=-4^{2 x+3}, \quad \text { for } f(-1)
$$

Yousef Sheikh

Numerade Educator

Problem 46

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
$$
f(x)=e^{x}, \text { for } f(3)
$$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 47

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
$$
f(x)=-2 e^{x-1}, \text { for } f(-1)
$$

Yousef Sheikh

Numerade Educator

Problem 48

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
$$
f(x)=2.7(4)^{-x+1}+1.5, \text { for } f(-2)
$$

H M

Numerade Educator

Problem 49

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
$$
f(x)=1.2 e^{2 x}-0.3, \text { for } f(3)
$$

Yousef Sheikh

Numerade Educator

Problem 50

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
$$
f(x)=-\frac{3}{2}(3)^{-x}+\frac{3}{2}, \text { for } f(2)
$$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 51

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.
$$
(0,3) \text { and }(3,375)
$$

Yousef Sheikh

Numerade Educator

Problem 52

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.
$$
(3,222.62) \text { and }(10,77.456)
$$

Allison Knapp

Numerade Educator

Problem 53

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.
$$
(20,29.495) \text { and }(150,730.89)
$$

Yousef Sheikh

Numerade Educator

Problem 54

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.
$$
(5,2.909) \text { and }(13,0.005)
$$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 55

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.
$$
(11,310.035) \text { and }(25,356.3652)
$$

Yousef Sheikh

Numerade Educator

Problem 56

The annual percentage yield (APY) of an investment account is actual interest rate earned on a compounding account. It is based on a compounding period of one year. Show that the APY of an account that compounds monthly can be found with the formula APY $=\left(1+\frac{r}{12}\right)^{12}-1$

K B

Numerade Educator

Problem 57

Repeat the previous exercise to find the formula for the APY of an account that compounds daily. Use the results from this and the previous exercise to develop a function $I(n)$ for the APY of any account that compounds $n$ times per year.

Joanna Quigley

Numerade Educator

Problem 58

Recall that an exponential function written in the form $f(x)=a \cdot b^{x}$ such that $a$ and $b$ are positive numbers and $b \neq 1 .$ Any positive number $b$ can be written as $b=e^{n}$ for some value of $n .$ Use this fact to rewrite the formula for an exponential function that uses the number $e$ as a base.

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 59

In an exponential decay function, the base of the exponent is a value between 0 and $1 .$ Thus, for some number $b>1$ the exponential decay function can be written as $f(x)=a \cdot\left(\frac{1}{b}\right)^{x}$ . Use this formula, along with the fact that $b=e^{n},$ to show that an exponential decay function takes the form $f(x)=a(e)^{-n x}$ for some positive number $n$

Yousef Sheikh

Numerade Educator

Problem 60

The formula for the amount $A$ in an investment account with a nominal interest rate $r$ any time $t$ is given by $A(t)=a(e)^{r t},$ where $a$ is the amount of principal initially deposited into an account that compounds continuously. Prove that the percentage of interest earned to principal at any time $t$ can be calculated with the formula $I(t)=e^{r t}-1$

K B

Numerade Educator

Problem 61

The fox population in a certain region has an annual growth rate of 9$\%$ per year. In the year $2012,$ there were $23,900$ fox counted in the area. What is the fox population predicted to be in the year 2020$?$

Yousef Sheikh

Numerade Educator

Problem 62

A scientist begins with 100 milligrams of a radioactive substance that decays exponentially. After 35 hours, 50 $\mathrm{mg}$ of the substance remains. How many milligrams will remain after 54 hours?

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 63

In the year $1985,$ a house was valued at $\$ 110,000 .$ By the year $2005,$ the valueciated to $\$ 145,000$ . What was the annual growth rate between 1985 and 2005$?$ Assume that the valued to grow by the same percentage. What was the value of the house in the year 2010$?$

Yousef Sheikh

Numerade Educator

Problem 64

A car was valued at $\$ 38,000$ in the year $2007 .$ By 2013 , the value had depreciated to $\$ 11,000$ If the car's value continues to drop by the same percentage, what will it be worth by 2017$?$

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 65

Jamal wants to save $\$ 54,000$ for a down payment on a home. How much will he need in an account with 8.2$\%$ APR, compounding daily, in order to reach his goal in 5 years?

Yousef Sheikh

Numerade Educator

Problem 66

Kyoko has $\$ 10,000$ that she wants to invest. Her bank has several investment accounts to choose from, all compounding daily. Her goal is to have $\$ 15,000$ by the time she finishes graduate school in 6 years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal? (Hint: solve the compound interest formula for the interest rate.)

Margaret Hubacher

Margaret Hubacher

Numerade Educator

Problem 67

Alyssa opened a retirement account with 7.25$\%$ APR in the year $2000 .$ Her initial deposit was $\$ 13,500 .$ How much will the account be worth in 2025 if interest compounds monthly? How much more would she make if interest compounded continuously?

Yousef Sheikh

Numerade Educator

Problem 68

An investment account with an annual interest rate of 7% was opened with an initial deposit of $4,000 Compare the values of the account after 9 years when the interest is compounded annually, quarterly, monthly, and continuously.

Pronoy Sinha

Numerade Educator