4.E: Exponential and Logarithmic Functions (Exercises)

4.1: Exponential Functions

When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions, which model this kind of rapid growth.

Verbal

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

Answer:: Linear functions have a constant rate of change. Exponential functions increase based on a percent of the original.

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.

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.

Answer:: When interest is compounded, the percentage of interest earned to principal ends up being greater than the annual percentage rate for the investment account. Thus, the annual percentage rate does not necessarily correspond to the real interest earned, which is the very definition of nominal.

Algebraic

For the following exercises, identify whether the statement represents an exponential function. Explain.

4) The average annual population increase of a pack of wolves is $25$.

5) A population of bacteria decreases by a factor of $\frac{1}{8}$ every $24$ hours.

Answer:: exponential; the population decreases by a proportional rate.

6) The value of a coin collection has increased by $3.25\%$annually over the last $20$ years.

7) For each training session, a personal trainer charges his clients $\$5$ less than the previous training session.

Answer:: not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function.

8) The height of a projectile at time $t$ is represented by the function $h(t)= -4.9t^2 + 18t + 40$

For the following exercises, consider this scenario: For each yearthe population of a forest of trees is represented by the functionIn a neighboring forest, the population of the same type of tree is represented by the function(Round answers to the nearest whole number.)

9) Which forest’s population is growing at a faster rate?

Answer:: The forest represented by the function

10) Which forest had a greater number of trees initially? By how many?

11) Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after $20$ years? By how many?

Answer:: Afteryears, forest A will have $43$more trees than forest B.

12) Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after $100$years? By how many?

13) 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?

Answer:: Answers will vary. Sample response: For a number of years, the population of forest A will increasingly exceed forest B, but because forest B actually grows at a faster rate, the population will eventually become larger than forest A and will remain that way as long as the population growth models hold. Some factors that might influence the long-term validity of the exponential growth model are drought, an epidemic that culls the population, and other environmental and biological factors.

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.

14)

15)

Answer:: exponential growth; The growth factor, $1.06$ is greater than

16) $y=16.5(1.025)^{\frac{1}{x}}$

17) $y=11,701(0.97)^t$

Answer:: exponential decay; The decay factor, is between $0$ and $1$

For the following exercises, find the formula for an exponential function that passes through the two points given.

18) and

19) $(0,2000)$and

Answer:: $f(x)=2000(0.1)^x$

20) $\left (−1,\frac{3}{2} \right )$ and $(3,24)$

21) \(and

Answer:: $f(x)=\left ( \frac{1}{6} \right )^{-\frac{3}{5}} \left ( \frac{1}{6} \right )^{\frac{x}{5}}\approx 2.93 (0.699)^x$

22) $(3,1)$ and $(5,4)$

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.

23)

	1	2	3	4
	70	40	10	-20

Answer:: Linear

24)

	1	2	3	4
	70	49	34.3	24.01

25)

	1	2	3	4
	80	61	42.9	25.61

Answer:: Neither

26)

	1	2	3	4
	10	20	40	80

27)

	1	2	3	4
	-3.25	2	7.25	12.5

Answer:: Linear

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

28) 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?

29) What was the initial deposit made to the account in the previous exercise?

Answer:: $\$10,250$

30) How many years had the account from the previous exercise been accumulating interest?

31) An account is opened with an initial deposit of $\$6,500$ and earnsinterest compounded semi-annually. What will the account be worth in $20$years?

Answer:: $\$13,268.58$

32) How much more would the account in the previous exercise have been worth if the interest were compounding weekly?

33) Solve the compound interest formula for the principal,.

Answer:: $P=A(t)\cdot \left (1+ \frac{r}{n} \right )^{-nt}$

34) Use the formula found in the previous exercise to calculate the initial deposit of an account that is worthafter earninginterest compounded monthly foryears. (Round to the nearest dollar.)

35) How much more would the account in the previous two exercises be worth if it were earning interest for $5$more years?

Answer:: $\$4,572.56$

36) Use properties of rational exponents to solve the compound interest formula for the interest rate,

37) Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $\$9,000$ and was worth $\$13,373.53$ after $10$ years.

Answer:: $4\%$

38) 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.

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.

39) $y=3742(e)^{0.75t}$

Answer:: continuous growth; the growth rate is greater than

40) $y=150(e)^{\frac{3.25}{t}}$

41) $y=2.25(e)^{-2t}$

Answer:: continuous decay; the growth rate is less than

42) Suppose an investment account is opened with an initial deposit ofearninginterest compounded continuously. How much will the account be worth after $30$years?

43) How much less would the account from Exercise 42 be worth afteryears if it were compounded monthly instead?

Answer:: $\$669.42$

Numeric

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.

44) $f(x)=2(5)^x$ for $f(-3)$

45) $f(x)=-4^{2x+3}$ for $f(-1)$

Answer:: $f(-1)=-4$

46) $f(x)=e^x$, for $f(3)$

47) $f(x)=-2e^{x-1}$, for $f(-1)$

Answer:: $f(-1)\approx -0.2707$

48) $f(x)=2.7(4)^{-x+1}+1.5$, for $f(-2)$

49) $f(x)=1.2e^{2x}-0.3$, for $f(3)$

Answer:: $f(3)\approx 483.8146$

50) $f(x)=-\frac{3}{2}(3)^{-x}+\frac{3}{2}$, for $f(2)$

Technology

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.

51) $(0,3)$ and $(3,375)$

Answer:: $y=3\cdot 5^x$

52) $(3,222.62)$ and $(10,77.456)$

53) $(20,29.495)$ and $(150,730.89)$

Answer:: $y\approx 18\cdot 1.025^x$

54) $(5,2.909)$ and $(13,0.005)$

55) ((11,310.035)\) and $(25,356.3652)$

Answer:: $y\approx 0.2\cdot 1.95^x$

Extensions

56) The annual percentage yield (APY) of an investment account is a representation of the 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$APY=\left (1+\frac{r}{12} \right )^{12}-1$

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.

Answer:: $\begin{align*} APY &= \frac{A(t)-a}{a}\\ &= \frac{a\left ( 1+\frac{r}{365} \right )^{365(1)}-a}{a}\\ &= \frac{a\left [\left ( 1+\frac{r}{365} \right )^{365}-1 \right ]}{a}\\ &= \left ( 1+\frac{r}{365} \right )^{365}-1 \end{align*}$; $I(n)=\left ( 1+\frac{r}{n} \right )^n - 1$

58) Recall that an exponential function is any equation written in the form $f(x)=a\cdot b^x$such that $a$ and $b$ are positive numbers andAny 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.

59) In an exponential decay function, the base of the exponent is a value between $0$ and $1$. Thus, for some numberthe 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$f(x)=a\cdot (e)^{-nx}$ for some positive number $n$.

Answer:

Let $f$ be the exponential decay function$f(x)=a\cdot \left (\frac{1}{b} \right )^x$ such that $b>1$. Then for some number $n>0$,

$\begin{align*} f(x) &= a\cdot \left (\frac{1}{b} \right )^x \\ &= a \left (b^{-1} \right )^x\\ &= a\left ( (e^n)^{-1} \right )^x\\ &= a\left ( e^{-n} \right )^x\\ &= a(e)^{-nx} \end{align*}$

60) The formula for the amount $A$in an investment account with a nominal interest rate $r$at any time $t$is given by $A(t)=a(e)^{rt}$, 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 $I(t)= e^{rt} - 1$

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?

Answer:: $47,622$ fox

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

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

Answer:: $1.39\%$; $\$155,368.09$

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?

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

Answer:: $\$35,838.76$

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.)

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?

Answer:: $\$82,247.78$; $\$449.75$

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.

4.2: Graphs of Exponential Functions

As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool.

Verbal

1) What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

Answer:: An asymptote is a line that the graph of a function approaches, as $x$ either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small.

2) What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

Algebraic

3) The graph of $f(x) = 3^x$is reflected about the $y$-axis and stretched vertically by a factor of $4$What is the equation of the new function, $g(x)$.State its $y$-intercept, domain, and range.

Answer:: $g(x)=4(3)^{-x}$ $(0,4)$Domain: all real numbers; Range: all real numbers greater than $0$.

4) The graph of $f(x)=\left ( \frac{1}{2} \right )^{-x}$is reflected about the $y$-axis and compressed vertically by a factor of $\left ( \frac{1}{5} \right )$What is the equation of the new function, $g(x)$State its $y$-intercept, domain, and range.

5) The graph of $f(x)=10^x$ is reflected about the $x$-axis and shifted upward $7$ units. What is the equation of the new function, $g(x)$?State its $y$-intercept, domain, and range.

Answer:: $g(x)=-10^x + 7$; $y$-intercept: $(0,6)$;Domain: all real numbers; Range: all real numbers less than $7$.

6) The graph of $f(x)=(1.68)^x$ is shifted right $3$ units, stretched vertically by a factor ofreflected about the $x$-axis, and then shifted downward $3$ units. What is the equation of the new function, $g(x)$? State its $y$-intercept (to the nearest thousandth), domain, and range.

7) The graph of $f(x)=2\left ( \frac{1}{4} \right )^{x-20}$ is shifted left $2$ units, stretched vertically by a factor of $4$reflected about the $x$-axis, and then shifted downward $4$ units. What is the equation of the new function, $g(x)$?State its y-intercept, domain, and range.

Answer:: $g(x)=2\left ( \frac{1}{4} \right )^x$; $y$-intercept: $(0,2)$;Domain: all real numbers; Range: all real numbers greater than $0$.

Graphical

For the following exercises, graph the function and its reflection about the $y$-axis on the same axes, and give the $y$-intercept.

8) $f(x)=3\left ( \frac{1}{2} \right )^x$

9) $g(x)=-2(0.25)^x$

Answer:

$CNX_PreCalc_Figure_04_02_202.jpg$

$y$-intercept: $(0,-2)$

10) $h(x)=6(1.75)^{-x}$

For the following exercises, graph each set of functions on the same axes.

11) $f(x)=3\left ( \frac{1}{4} \right )^x, g(x)=3(2)^x, h(x)=3(4)^x$

Answer:: $CNX_PreCalc_Figure_04_02_204.jpg$

12) $f(x)=\frac{1}{4}(3)^x, g(x)=2(3)^x, h(x)=4(3)^x$

For the following exercises, match each function with one of the graphs in Figure below.

$CNX_PreCalc_Figure_04_02_206.jpg$

13) $f(x)=2(0.69)^x$

Answer:: B

14) $f(x)=2(1.28)^x$

15) $f(x)=2(0.81)^x$

Answer:: A

16) $f(x)=4(1.28)^x$

17) $f(x)=2(1.59)^x$

Answer:: E

18) $f(x)=4(0.69)^x$

For the following exercises, use the graphs shown in Figure below. All have the form $f(x)=ab^x$.

$CNX_PreCalc_Figure_04_02_207.jpg$

19) Which graph has the largest value for $b$

Answer:: D

20) Which graph has the smallest value for $b$?

21) Which graph has the largest value for $a$?

Answer:: C

22) Which graph has the smallest value for $a$?

For the following exercises, graph the function and its reflection about the $x$-axis on the same axes.

23) $f(x)=\frac{1}{2}(4)^x$

Answer:: $CNX_PreCalc_Figure_04_02_208.jpg$

24) $f(x)=3(0.75)^x-1$

25) $f(x)=-4(2)^x+2$

Answer:: $CNX_PreCalc_Figure_04_02_210.jpg$

For the following exercises, graph the transformation of $f(x)=2^x$ Give the horizontal asymptote, the domain, and the range.

26) $f(x)=2^{-x}$

27) $h(x)=2^x+3$

Answer:

$CNX_PreCalc_Figure_04_02_212.jpg$

Horizontal asymptote: $h(x)=3$ Domain: all real numbers; Range: all real numbers strictly greater than $3$.

28) $f(x)=2^{x-2}$

For the following exercises, describe the end behavior of the graphs of the functions.

29) $f(x)=-5(4)^x-1$

Answer:

As $x\rightarrow \infty , f(x)\rightarrow -\infty$

As $x\rightarrow -\infty , f(x)\rightarrow -1$

30) $f(x)=3\left ( \frac{1}{2} \right )^x-2$

31) $f(x)=3(4)^{-x}+2$

Answer:

As $x\rightarrow \infty , f(x)\rightarrow 2$

As $x\rightarrow -\infty , f(x)\rightarrow \infty$

For the following exercises, start with the graph of $f(x)=4^x$ Then write a function that results from the given transformation.

32) Shift $f(x)$ $4$ units upward

33) Shift $f(x)$ $3$ units downward

Answer:: $f(x)=4^x-3$

34) Shift $f(x)$ $2$ units left

35) Shift $f(x)$ $5$ units right

Answer:: $f(x)=4^{x-5}$

36) Reflect $f(x)$ about the $x$-axis

37) Reflect $f(x)$ about the $y$-axis

Answer:: $f(x)=4^{-x}$

For the following exercises, each graph is a transformation of $f(x)=2^x$ Write an equation describing the transformation.

38)

$CNX_PreCalc_Figure_04_02_214.jpg$

39)

$CNX_PreCalc_Figure_04_02_215.jpg$

Answer:: $y=-2^x+3$

40)

$CNX_PreCalc_Figure_04_02_216.jpg$

For the following exercises, find an exponential equation for the graph.

41)

$CNX_PreCalc_Figure_04_02_217.jpg$

Answer:: $y=-2(3)^x+7$

42)

$CNX_PreCalc_Figure_04_02_218.jpg$

Numeric

For the following exercises, evaluate the exponential functions for the indicated value of $x$.

43) $g(x)=\frac{1}{3}(7)^{x-2}$ for $g(6)$.

Answer:: $g(6)=800+\frac{1}{3}\approx 800.3333$

44) $f(x)=4(2)^{x-1}-2$ for $f(5)$.

45) $h(x)=-\frac{1}{2}\left ( \frac{1}{2} \right )^x+6$ for $h(-7)$.

Answer:: $h(-7)=-58$

Technology

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.

46) $-50=\left ( \frac{1}{2} \right )^{-x}$

47) $116=\left ( \frac{1}{4} \right )\left ( \frac{1}{8} \right )^x$

Answer:: $x\approx -2.953$

48) $12=2(3)^x+1$

49) $5=3\left ( \frac{1}{2} \right )^{x-1}-2$

Answer:: $x\approx -0.222$

50) $-30=-4(2)^{x+2}+2$

Extensions

51) Explore and discuss the graphs of$F(x)=(b)^x$ and $G(x)=\left ( \frac{1}{b} \right )^x$. Then make a conjecture about the relationship between the graphs of the functions $b^x$ and $\left ( \frac{1}{b} \right )^x$ for any real number

Answer:: The graph of $G(x)=\left ( \frac{1}{b} \right )^x$ is the refelction about the $y$-axis of the graph of $F(x)=(b)^x$; For any real number $b>0$ and function $f(x)=(b)^x$ the graph of $\left ( \frac{1}{b} \right )^x$ is the the reflection about the $y$-axis, $F(-x)$.

52) Prove the conjecture made in the previous exercise.

53) Explore and discuss the graphs of $f(x) = 4^x$, $g(x)=4^{x-2}$, and $h(x)=\left ( \frac{1}{16} \right )4^x$ Then make a conjecture about the relationship between the graphs of the functions $b^x$ and $\left ( \frac{1}{b^n} \right )b^x$ for any real number $n$ and real number $b>0$.

Answer:: The graphs of $g(x)$ and $h(x)$ are the same and are a horizontal shift to the right of the graph of $f(x)$; For any real number $n$, real number $b>0$, and function $f(x)=b^x$ the graph of $\left ( \frac{1}{b^n} \right )b^x$ is the horizontal shift $f(x-n)$.

54) Prove the conjecture made in the previous exercise.

4.3: Logarithmic Functions

The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.

Verbal

1) What is a base $b$ logarithm? Discuss the meaning by interpreting each part of the equivalent equations $b^y=x$ and $\log _bx=y$ for $b>0, b\neq 1$

Answer: A logarithm is an exponent. Specifically, it is the exponent to which a base $b$ is raised to produce a given value. In the expressions given, the base $b$ has the same value. The exponent, $y$in the expression $b^y$ can also be written as the logarithm, $\log _bx=y$and the value of $x$ is the result of raising $b$ to the power of $y$.

2) How is the logarithmic function $f(x)=\log _bx$ related to the exponential function $g(x)=b^x$? What is the result of composing these two functions?

3) How can the logarithmic equation $\log _bx=y$ be solved for $x$ using the properties of exponents?

Answer: Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation $b^y = x$ and then properties of exponents can be applied to solve for $x$

4) Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base $b$ and how does the notation differ?

5) Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base $b$ and how does the notation differ?

Answer: The natural logarithm is a special case of the logarithm with base $b$ in that the natural log always has base $e$.Rather than notating the natural logarithm as $\log_{e}(x)$the notation used is $\ln (x)$

Algebraic

For the following exercises, rewrite each equation in exponential form.

6) $\log_{4}(q)=m$

7) $\log_{a}(b)=c$

Answer: $a^c=b$

8) $\log_{16}(y)=x$

9) $\log_{x}(64)=y$

Answer: $x^y=64$

10) $\log_{y}(x)=-11$

11) $\log_{15}(a)=b$

Answer: $15^b=a$

12) $\log_{y}(137)=x$

13) $\log_{13}(142)=a$

Answer: $13^a=142$

14) $\log(v)=t$

15) $\ln(w)=n$

Answer: $e^n=w$

For the following exercises, rewrite each equation in logarithmic form.

16) $4^x=y$

17) $c^d=k$

Answer: $\log_{c}(k)=d$

18) $m^{-7}=n$

19) $19^x=y$

Answer: $\log_{19}(y)=x$

20) $x^{-\frac{10}{13}}=y$

21) $n^4 = 103$

Answer: $\log_{n}(103)=4$

22) $\left ( \dfrac{7}{5} \right )^m=n$

23) $y^x=\dfrac{39}{100}$

Answer: $\log_{y}\left ( \dfrac{39}{100} \right )=x$

24) $10^a=b$

25) $e^k=h$

Answer: $\ln(w)=n$

For the following exercises, solve for $x$ by converting the logarithmic equation to exponential form.

26) $\log_{3}(x)=2$

27) $\log_{2}(x)=-3$

Answer: $x=2^{-3}=\dfrac{1}{8}$

28) $\log_{5}(x)=2$

29) $\log_{3}(x)=3$

Answer: $x = 3^3 = 27$

30) $\log_{2}(x)=6$

31) $\log_{9}(x)=\dfrac{1}{2}$

Answer: $x=9^{\frac{1}{2}}=3$

32) $\log_{18}(x)=2$

33) $\log_{6}(x)=-3$

Answer: $x=6^{-3}=\dfrac{1}{216}$

34) $\log (x)=3$

35) $\ln(x)=2$

Answer: $x=e^2$

For the following exercises, use the definition of common and natural logarithms to simplify.

36) $\log (100^8)$

37) $10^{\log (32)}$

Answer: $32$

38) $2\log (.0001)$

39) $e^{\ln (1.06)}$

Answer: $1.06$

40) $\ln (e^{-5.03})$

41) $e^{\ln (10.125)}+4$

Answer: $14.125$

Numeric

For the following exercises, evaluate the base $b$ logarithmic expression without using a calculator.

42) $\log _3\left ( \frac{1}{27} \right )$

43) $\log _6(\sqrt{6})$

Answer: $\dfrac{1}{2} $

44) $\log _2\left ( \frac{1}{8} \right )+4$

45) $6\log _8(4)$

Answer: $4$

For the following exercises, evaluate the common logarithmic expression without using a calculator.

46) $\log (10,000)$

47) $\log (0.001)$

Answer: $-3$

48) $\log (1)+7$

49) $2\log (100^{-3})$

Answer: $-12$

For the following exercises, evaluate the natural logarithmic expression without using a calculator.

50) $\ln \left ( e^{\frac{1}{3}} \right )$

51) $\ln (1)$

Answer: $0$

52) $\ln \left ( e^{-0.225} \right )-3$

53) $25\ln \left ( e^{\frac{2}{5}} \right )$

Answer: $10$

Technology

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

54) $\log (0.04)$

55) $\ln (15)$

Answer: $2.708$

56) $\ln \left ( {\frac{4}{5}} \right )$

57) $\log (\sqrt{2})$

Answer: $0.151$

58) $\ln (\sqrt{2})$

Extensions

59) Is $x=0$in the domain of the function $f(x)=\log x$? If so, what is the value of the function when $x=0$? Verify the result.

Answer: No, the function has no defined value for $x=0$ To verify, suppose $x=0$ is in the domain of the function $f(x)=\log (x)$ Then there is some number $n$ such that $n=\log(0)$ Rewriting as an exponential equation gives: $10^n=0$ which is impossible since no such real number $n$ exists. Therefore, $x=0$ is not the domain of the function $f(x)=\log (x)$.

60) Is $f(x)=0$ in the range of the function $f(x)=\log (x)$If so, for what value of $x$?Verify the result.

61) Is there a number $x$ such that $\ln x = 2$? If so, what is that number? Verify the result.

Answer: Yes. Suppose there exists a real number $x$ such that $\ln x = 2$Rewriting as an exponential equation gives $x=e^2$ which is a real number. To verify, let $x=e^2$Then, by definition, $\ln (x)=\ln \left ( e^2 \right ) = 2$

62) Is the following true: $\frac{\log _3(27)}{\log _4\left ( \frac{1}{64} \right )}=-1$ $\frac{\log_{3} (27)}{\log_{4} (\frac{1}{64})} = −1 ?$ Verify the result.

63) Is the following true:$\frac{\ln (e^{1.725})}{\ln (1)}=1.725$ Verify the result.

Answer: No; $\ln (1) =0$, so $\frac{\ln (e^{1.725})}{\ln (1)}=1.725$ is undefined.

Real-World Applications

64) The exposure index $EI$ for a $35$ millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation $EI=\log _2\left ( \frac{f^2}{t} \right )$ where $f$ is the “f-stop” setting on the camera, and $t$ is the exposure time in seconds. Suppose the f-stop setting is $8$ and the desired exposure time is $2$seconds. What will the resulting exposure index be?

65) Refer to the previous exercise. Suppose the light meter on a camera indicates an $EI$ of and the desired exposure time is $16$ seconds. What should the f-stop setting be?

Answer: $2$

66) The intensity levels $I$ of two earthquakes measured on a seismograph can be compared by the formula $\log \left ( \frac{I_1}{I_2} \right )=M_1-M_2$where $M$ is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude $6.1$ hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of $9.0$. How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

4.4: Graphs of Logarithmic Functions

In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.

Verbal

1) The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?

Answer: Since the functions are inverses, their graphs are mirror images about the line $y-x$So for every point $(a,b)$ on the graph of a logarithmic function, there is a corresponding point $(b,a)$ on the graph of its inverse exponential function.

2) What type(s) of translation(s), if any, affect the range of a logarithmic function?

3) What type(s) of translation(s), if any, affect the domain of a logarithmic function?

Answer: Shifting the function right or left and reflecting the function about the $y$-axis will affect its domain.

4) Consider the general logarithmic function $f(x)=\log _b(x)$Why can’t $x$ be zero?

5) Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

Answer: No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.

Algebraic

For the following exercises, state the domain and range of the function.

6) $f(x)=\log _3(x+4)$

7) $h(x)=\ln \left ( \dfrac{1}{2}-x \right )$

Answer: Domain: $\left ( -\infty , \dfrac{1}{2} \right )$; Range: $(-\infty , \infty )$

8) $g(x)=\log _5(2x+9)-2$

9) $h(x)=\ln (4x+17)-5$

Answer: Domain: $\left ( -\dfrac{17}{4}, \infty \right )$; Range: $(-\infty , \infty )$

10) $f(x)=\log _2 (12-3x)-3$

For the following exercises, state the domain and the vertical asymptote of the function.

11) $f(x)=\log _b (x-5)$

Answer: Domain: $(5, \infty )$; Vertical asymptote: $x=5$

12) $g(x)=\ln (3-x)$

13) $f(x)=\log (3x+1)$

Answer: Domain: $\left ( -\dfrac{1}{3}, \infty \right )$; Vertical asymptote: $x=-\dfrac{1}{3}$

14) $f(x)=3\log (-x)+2$

15) $g(x)=-\ln (3x+9)-7$

Answer: Domain: $(-3, \infty )$; Vertical asymptote: $x=-3$

For the following exercises, state the domain, vertical asymptote, and end behavior of the function.

16) $f(x)=\ln (2-x)$

17) $f(x)=\log \left ( x-\dfrac{3}{7} \right )$

Answer

Domain: $\left ( \dfrac{3}{7},\infty \right )$

Vertical asymptote: $x=\dfrac{3}{7}$

End behavior: as $x\rightarrow \left (\dfrac{3}{7} \right )^+$, $f(x)\rightarrow -\infty$ and as $x\rightarrow \infty ,f(x)\rightarrow \infty$

18) $h(x)=-\log (3x-4)+3$

19) $g(x)=\ln (2x+6)-5$

Answer

Domain: $\left ( -3,\infty \right )$

Vertical asymptote: $x=-3$

End behavior: as $x\rightarrow -3^+$, $f(x)\rightarrow -\infty$ and as $x\rightarrow \infty ,f(x)\rightarrow \infty$

20) $f(x)=\log_3 (15-5x)+6$

For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.

21) $h(x)=\log_4 (x-1)+1$

Answer

Domain: $\left (1,\infty \right )$

Range: $-\infty , \infty $

Vertical asymptote: $x=1$

$x$-intercept: $\left ( \dfrac{5}{4},0\right )$

$y$-intercept: DNE

22) $f(x)=\log (5x+10)+3$

23) $g(x)=\ln (-x)-2$

Answer

Domain: $\left (-\infty ,0 \right )$

Range: $-\infty , \infty $

Vertical asymptote: $x=0$

$x$-intercept: $\left ( -e^2,0 \right )$

$y$-intercept: DNE

24) $f(x)=\log_2 (x+2)-5$

25) $h(x)=3\ln (x)-9$

Answer

Domain: $\left (0,\infty \right )$

Range: $-\infty , \infty $

Vertical asymptote: $x=0$

$x$-intercept: $\left ( e^3,0 \right )$

$y$-intercept: DNE

Graphical

For the following exercises, match each function in Figure below with the letter corresponding to its graph.

$Ex. 4.4.26.png$

26) $d(x)=\log (x)$

27) $f(x)=\ln (x)$

Answer: $B$

28) $g(x)=\log_2 (x)$

29) $h(x)=\log_5 (x)$

Answer: $C$

30) $j(x)=\log_{25} (x)$

For the following exercises, match each function in Figure with the letter corresponding to its graph.

$Ex 4.4.31.png$

31) $f(x)=\log_{\frac{1}{3}} (x)$

Answer: $B$

32) $g(x)=\log_2 (x)$

33) $h(x)=\log_{\frac{3}{4}} (x)$

Answer: $C$

For the following exercises, sketch the graphs of each pair of functions on the same axis.

34) $f(x)=\log (x)$ and $g(x)=10^x$

35) $f(x)=e^x$ and $g(x)=\ln (x)$

Answer: $Ex 4.4.35.png$

For the following exercises, match each function in Figure with the letter corresponding to its graph.

36) $f(x)=\log _4(-x+2)$

37) $g(x)=-\log _4(x+2)$

Answer: $C$

38) $h(x)=\log _4(x+2)$

For the following exercises, sketch the graph of the indicated function.

39) $f(x)=\log _2(x+2)$

Answer: $Ex 4.4.39.png$

40) $f(x)=2\log (x)$

41) $f(x)=\ln (-x)$

Answer: $Ex 4.4.41.png$

42) $g(x)=\log (4x+16)+4$

43) $g(x)=\log (6-3x)+1$

Answer: $Ex 4.4.43.png$

44) $h(x)=-\dfrac{1}{2}\log (x+1)-3$

For the following exercises, write a logarithmic equation corresponding to the graph shown.

45) Use $y=\log _2(x)$as the parent function.

$CNX_PreCalc_Figure_04_02_214.jpg$

Answer: $f(x)=\log _2(-(x-1))$

46) Use $f(x)=\log _3(x)$ as the parent function.

$CNX_PreCalc_Figure_04_04_215.jpg$

47) Use $f(x)=\log _4(x)$ as the parent function.

$CNX_PreCalc_Figure_04_04_216.jpg$

Answer: $f(x)=3\log _4(x+2)$

48) Use $f(x)=\log _5(x)$ as the parent function.

$CNX_PreCalc_Figure_04_04_217.jpg$

Technology

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

49) $\log (x-1)+2=\ln (x-1)+2$

Answer: $x=2$

50) $\log (2x-3)+2=-\log (2x-3)+5$

51) $\ln (x-2)+2=-\ln (x+1)$

Answer: $x\approx 2.303$

52) $2\ln (5x+1)=\dfrac{1}{2}\ln (-5x)+1$

53) $\dfrac{1}{3}\log (1-x)=\log (x+1)+\dfrac{1}{3}$

Answer: $x\approx -0.472$

Extensions

54) Let $b$ be any positive real number such that $b\neq 1$What must $\log _b 1 $be equal to? Verify the result.

55) Explore and discuss the graphs of $f(x)=\log_{\frac{1}{2}}(x)$ and $g(x)=-\log _2(x)$Make a conjecture based on the result.

Answer

The graphs of $f(x)=\log_{\frac{1}{2}}(x)$ and $g(x)=-\log _2(x)$ appear to be the same;

Conjecture: for any positive base $b\neq 1$, $\log_{b}(x)=\log_{\frac{1}{b}}(x)$

56) Prove the conjecture made in the previous exercise.

57) What is the domain of the function $f(x)=\ln \left (\frac{x+2}{x-4} \right )$Discuss the result.

Answer

Recall that the argument of a logarithmic function must be positive, so we determine where $\frac{x+2}{x-4}> 0$. From the graph of the function $f(x)=\frac{x+2}{x-4}$ $,$ note that the graph lies above the $x$-axis on the interval $(-\infty ,-2)$ and again to the right of the vertical asymptote, that is $(4,\infty )$ Therefore, the domain is $(-\infty ,-2)\cup (4,\infty )$

$CNX_Precalc_Figure_04_04_219.jpg$

58) Use properties of exponents to find the $x$-intercepts of the function $f(x)=\log \left ( x^2+4x+4 \right )$ algebraically. Show the steps for solving, and then verify the result by graphing the function.

4.5: Logarithmic Properties

Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here.

Verbal

1) How does the power rule for logarithms help when solving logarithms with the form $\log _b(\sqrt[n]{x})$

Answer: Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus, $\log _b \left ( x^{\frac{1}{n}} \right ) = \dfrac{1}{n}\log_{b}(x)$.

2) What does the change-of-base formula do? Why is it useful when using a calculator?

Algebraic

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

3) $\log _b (7x\cdot 2y)$

Answer: $\log _b (2)+\log _b (7)+\log _b (x)+\log _b (y)$

4) $\ln (3ab\cdot 5c)$

5) $\log_b \left ( \dfrac{13}{17} \right )$

Answer: $\log _b (13)-\log _b (17)$

6) $\log_4 \left ( \dfrac{\frac{x}{z}}{w} \right )$

7) $\ln \left ( \dfrac{1}{4^k} \right )$

Answer: $-k\ln(4)$

8) $\log _2 (y^x)$

For the following exercises, condense to a single logarithm if possible.

9) $\ln (7)+\ln (x)+\ln (y)$

Answer: $\ln(7xy)$

10) $\log_3(2)+\log_3(a)+\log_3(11)+\log_3(b)$

11) $\log_b(28)-\log_b(7)$

Answer: $\log_b(4$

12) $\ln (a)-\ln (d)-\ln (c)$

13) $-\log_b\left ( \dfrac{1}{7} \right )$

Answer: $\log_b(7$

14) $\dfrac{1}{3}\ln(8)$

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

15) $\log \left ( \dfrac{x^{15}y^{13}}{z^{19}} \right )$

Answer: $15\log (x)+13\log (y)-19\log (z)$

16) $\ln \left ( \frac{a^{-2}}{b^{-4}c^{5}} \right )$

17) $\log \left ( \sqrt{x^3y^{-4}} \right )$

Answer: $\frac{3}{2}\log (x)-2\log (y)$

18) $\ln \left ( y\sqrt{\frac{y}{1-y}} \right )$

19) $\log \left ( x^2y^3 \sqrt[3]{x^2y^5} \right )$

Answer: $\dfrac{8}{3}\log (x)+\dfrac{14}{3}\log (y)$

For the following exercises, condense each expression to a single logarithm using the properties of logarithms.

20) $\log \left ( 2x^4 \right )+\log \left (3x^5 \right )$

21) $\ln \left ( 6x^9 \right )-\ln \left (3x^2 \right )$

Answer: $\ln \left ( 2x^7 \right )$

22) $2\log (x)+3\log (x+1)$

23) $\log (x)-\dfrac{1}{2}\log (y)+3\log (z)$

Answer: $\log \left ( \dfrac{xz^3}{\sqrt{y}} \right )$

24) $4\log _7(c)+\dfrac{\log _7(a)}{3}+\dfrac{\log _7(b)}{3}$

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.

25) $\log _7(15)$ to base $e$

Answer: $\log _7(15)=\dfrac{\ln (15)}{\ln (7)}$

26) $\log _14(55.875)$ to base $10$

For the following exercises, suppose $\log _5(6)=a$ and $\log _5(11)=b$ Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of $a$ and $b$ Show the steps for solving.

27) $\log _{11} (5)$

Answer: $\log _{11} (5)=\dfrac{\log_5 (5)}{\log_5 (11)}=\dfrac{1}{b}$

28) $\log _{6} (55)$

29) $\log _{11}\left (\dfrac{6}{11} \right )$

Answer: $\log _{11}\left (\dfrac{6}{11} \right )=\dfrac{\log _{11}\left (\dfrac{6}{11} \right )}{\log _{5}(11)}=\dfrac{\log _{5}(6)-\log _{5}(11)}{\log _{5}(11)}=\dfrac{a-b}{b}=\dfrac{a}{b}-1$

Numeric

For the following exercises, use properties of logarithms to evaluate without using a calculator.

30) $\log _3 \left ( \dfrac{1}{9} \right )-3\log _3 (3)$

31) $6\log _8 (2)+\dfrac{\log _8 (64)}{3\log _8 (4)}$

Answer: $3$

32) $2\log _9 (3)-4\log _9 (3)+\log _9 \left (\dfrac{1}{729} \right )$

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

33) \log _3 (22)

Answer: $2.81359$

34) \log _8 (65)

35) \log _6 (5.38)

Answer: $0.93913$

36) $\log _4 \left (\dfrac{15}{2} \right )$

37) $\log _{\frac{1}{2}} (4.7)$

Answer: $-2.23266$

Extensions

38) Use the product rule for logarithms to find all $x$ values such that $\log _{12} (2x+6)+\log _{12} (x+2)=2$Show the steps for solving.

39) Use the quotient rule for logarithms to find all $x$ values such that $\log _{6} (x+2)-\log _{6} (x-3)=1$Show the steps for solving.

Answer

Rewriting as an exponential equation and solving for $x$:

$\begin{align*}
6^1 &= \frac{x+2}{x-3}\\
0 &= \frac{x+2}{x-3}-6\\
0 &= \frac{x+2}{x-3}-\frac{6(x-3)}{(x-3)}\\
0 &= \frac{x+2-6x+18}{x-3}\\
0 &= \frac{x-4}{x-3}\\
x &= 4
\end{align*}$

Checking, we find that $\log _6(4+2)-\log _6(4-3)=\log _6(6)-\log _6(1)$ is defined, so $x=4$

40) Can the power property of logarithms be derived from the power property of exponents using the equation $b^x=m$If not, explain why. If so, show the derivation.

41) Prove that $\log_{b} (n) = \frac{1}{\log_{n}}$ for any positive integers $b > 1$ and $n > 1.$

4.6: Exponential and Logarithmic Equations

Uncontrolled population growth can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.

4.7: Exponential and Logarithmic Models

We have already explored some basic applications of exponential and logarithmic functions. In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling.

4.8: Fitting Exponential Models to Data

We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review.