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  • Beginning and Intermediate Algebra
  • Inverse, Exponential, and Logarithmic Functions

Beginning and Intermediate Algebra

Margaret L. Lial , John Hornsby

Chapter 12

Inverse, Exponential, and Logarithmic Functions - all with Video Answers

Educators

Section 5

Common and Natural Logarithms

00:16

Problem 1

What is the base in the expression $\log x ?$
A. $e$
B. 1
C. 10
D. $x$

Amy J.
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00:14

Problem 2

What is the base in the expression $\ln x ?$
A. $e$
B. 1
C. 10
D. $x$

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00:51

Problem 3

Since $10^{0}=1$ and $10^{1}=10,$ between what two consecutive integers is the value of $\log 6.3 ?$
A. 6 and 7
B. 10 and 11
C. 0 and $1$
D. $-1$ and 0

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00:56

Problem 4

Since $e^{1} \approx 2.718$ and $e^{2} \approx 7.389,$ between what two consecutive integers is the value of $\ln 6.3 ?$
A. 6 and 7
B. 2 and 3
C. 1 and 2
D. 0 and 1

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00:16

Problem 5

Without using a calculator, give the value of $\log 10^{31.6}$

Amy J.
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00:17

Problem 6

Without using a calculator, give the value of $\ln e^{\sqrt{3}}$.

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00:14

Problem 7

Find each logarithm. Give approximations to four decimal places.
$\log 43$

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00:13

Problem 8

Find each logarithm. Give approximations to four decimal places.
$\log 98$

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00:17

Problem 9

Find each logarithm. Give approximations to four decimal places.
$\log 328.4$

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00:13

Problem 10

Find each logarithm. Give approximations to four decimal places.
$\log 457.2$

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00:17

Problem 11

Find each logarithm. Give approximations to four decimal places.
$\log 0.0326$

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00:12

Problem 12

Find each logarithm. Give approximations to four decimal places.
$\log 0.1741$

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00:15

Problem 13

Find each logarithm. Give approximations to four decimal places.
$\log \left(4.76 \times 10^{9}\right)$

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00:14

Problem 14

Find each logarithm. Give approximations to four decimal places.
$\log \left(2.13 \times 10^{4}\right)$

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00:08

Problem 15

Find each logarithm. Give approximations to four decimal places.
$\ln 7.84$

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00:12

Problem 16

Find each logarithm. Give approximations to four decimal places.
$\ln 8.32$

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00:11

Problem 17

Find each logarithm. Give approximations to four decimal places.
$\ln 0.0556$

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00:09

Problem 18

Find each logarithm. Give approximations to four decimal places.
$\ln 0.0217$

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00:10

Problem 19

Find each logarithm. Give approximations to four decimal places.
$\ln 388.1$

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00:11

Problem 20

Find each logarithm. Give approximations to four decimal places.
$\ln 942.6$

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00:18

Problem 21

Find each logarithm. Give approximations to four decimal places.
$\ln \left(8.59 \times e^{2}\right)$

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00:12

Problem 22

Find each logarithm. Give approximations to four decimal places.
$\ln \left(7.46 \times e^{3}\right)$

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00:09

Problem 23

Find each logarithm. Give approximations to four decimal places.
$\ln 10$

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00:10

Problem 24

Find each logarithm. Give approximations to four decimal places.
$\log e$

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02:04

Problem 25

Use your calculator to find approximations of the following logarithms.
(a) $\log 356.8$
(b) $\log 35.68$
(c) $\log 3.568$
(d) Observe your answers and make a conjecture concerning the decimal values of the common logarithms of numbers greater than 1 that have the same digits.

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01:26

Problem 26

Let $k$ represent the number of letters in your last name.
(a) Use your calculator to find $\log k$
(b) Raise 10 to the power indicated by the number in part (a). What is your result?
(c) Use the concepts of Section 12.1 to explain why you obtained the answer you found
in part (b). Would it matter what number you used for $k$ to observe the same result?

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00:35

Problem 27

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Is the wetland a rich fen, a poor fen, or a bog?
$3.1 \times 10^{-5}$

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00:32

Problem 28

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Is the wetland a rich fen, a poor fen, or a bog?
$2.5 \times 10^{-5}$

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00:31

Problem 29

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Is the wetland a rich fen, a poor fen, or a bog?
$2.5 \times 10^{-2}$

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00:26

Problem 30

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Is the wetland a rich fen, a poor fen, or a bog?
$3.6 \times 10^{-2}$

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00:28

Problem 31

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Is the wetland a rich fen, a poor fen, or a bog?
$2.7 \times 10^{-7}$

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00:30

Problem 32

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Is the wetland a rich fen, a poor fen, or a bog?
$2.5 \times 10^{-7}$

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00:24

Problem 33

Find the pH of the substance with the given hydronium ion concentration.
Ammonia, $2.5 \times 10^{-12}$

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00:28

Problem 34

Find the pH of the substance with the given hydronium ion concentration.
Sodium bicarbonate, $4.0 \times 10^{-9}$

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00:28

Problem 35

Find the pH of the substance with the given hydronium ion concentration.
Grapes, $5.0 \times 10^{-5}$

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00:22

Problem 36

Find the pH of the substance with the given hydronium ion concentration.
Tuna, $1.3 \times 10^{-6}$

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00:34

Problem 37

Find the hydronium ion concentration of the substance with the given pH.
Human blood plasma, 7.4

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00:30

Problem 38

Find the hydronium ion concentration of the substance with the given pH.
Human gastric contents, 2.0

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00:31

Problem 39

Find the hydronium ion concentration of the substance with the given pH.
Spinach, 5.4

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00:26

Problem 40

Find the hydronium ion concentration of the substance with the given pH.
Spinach, 5.4

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01:33

Problem 41

Consumers can now enjoy movies at home in elaborate home-theater systems. Find the average decibel level
$$D=10 \log \left(\frac{I}{I_{0}}\right)$$
for each movie with the given intensity $I$
(a) Avatar; $5.012 \times 10^{10} \mathrm{I}_{0}$
(b) Iron Man 2; $10^{10} I_{0}$
(c) Clash of the Titans; $6,310,000,000 \mathrm{I}_{0}$

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01:19

Problem 42

The time $t$ in years for an amount increasing at a rate of $r$ (in decimal form) to double is given by
$$t(r)=\frac{\ln 2}{\ln (1+r)}$$
This is called doubling time. Find the doubling time to the nearest tenth for an investment at each interest rate.
(a) $2 \% \text { (or } 0.02)$
(b) $5 \% \text { (or } 0.05)$
(c) $8 \% \text { (or } 0.08)$

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01:19

Problem 43

The number of years, $N(r),$ since two independently evolving languages split off from a common ancestral language is approximated by
$$N(r)=-5000 \ln r$$
where $r$ is the percent of words (in decimal form) from the ancestral language common to both languages now. Find the number of years (to the nearest hundred years) since the split for each percent of common words.
(a) $85 \% \text { (or } 0.85)$
(b) $35 \% \text { (or } 0.35)$
(c) $10 \% \text { (or } 0.10)$

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00:45

Problem 44

The concentration of a drug injected into the bloodstream decreases with time. The intervals of time $T$ when the drug should be administered are given by
$$T=\frac{1}{k} \ln \frac{C_{2}}{C_{1}}$$
where $k$ is a constant determined by the drug in use, $C_{2}$ is the concentration at which the drug is harmful, and $C_{1}$ is the concentration below which the drug is ineffective. (Source: Horelick, Brindell and Sinan Koont, "Applications of Calculus to Medicine: Prescribing Safe and Effective Dosage," UMAP Module 202.) Thus, if $T=4,$ the drug should be administered every 4 hr. For a certain drug, $k=\frac{1}{3}, C_{2}=5,$ and $C_{1}=2 .$ How often should the drug be administered?

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02:08

Problem 45

The growth of outpatient surgeries as a percent of total surgeries at hospitals is approximated by
$$f(x)=-1317+304 \ln x$$
where $x$ is the number of years since $1900 .$ (Source: American Hospital Association.)
(a) What does this function predict for the percent of outpatient surgeries in $1998 ?$
(b) When did outpatient surgeries reach $50 \% ?$ (Hint: Substitute for $y,$ then write the equation in exponential form to solve it.)

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00:54

Problem 46

In the central Sierra Nevada of California, the percent of moisture that falls as snow rather than rain is approximated reasonably well by
$$f(x)=86.3 \ln x-680$$
where $x$ is the altitude in feet.
(a) What percent of the moisture at 5000 ft falls as snow?
(b) What percent at 7500 ft falls as snow?

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01:51

Problem 47

The cost-benefit equation
$$T=-0.642-189 \ln (1-p)$$
describes the approximate tax $T,$ in dollars per ton, that would result in a $p \%$ (in decimal form) reduction in carbon dioxide emissions.
(a) What tax will reduce emissions $25 \% ?$
(b) Explain why the equation is not valid for $p=0$ or $p=1$

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02:07

Problem 48

The age in years of a female blue whale of length $L$ in feet is approximated by $t=-2.57 \ln \left(\frac{87-L}{63}\right)$.
(a) How old is a female blue whale that measures $80 \mathrm{ft} ?$
(b) The equation that defines $t$ has domain $24<L<87 .$ Explain why.

Amy J.
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00:18

Problem 49

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places.
$\log _{3} 12$

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00:20

Problem 50

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places.
$\log _{\pi} e$

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00:18

Problem 51

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places.
$\log _{5} 3$

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00:18

Problem 52

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places.
$\log _{7} 4$

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00:21

Problem 53

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places.
$\log _{3} \sqrt{2}$

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00:30

Problem 54

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places.
$\log _{6} \sqrt[3]{5}$

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00:22

Problem 55

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places.
$\log _{\pi} e$

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00:18

Problem 56

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places.
$\log _{\pi} 10$

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00:16

Problem 57

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places.
$\log _{e} 12$

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01:35

Problem 58

To solve the equation $5^{x}=7,$ we must find the exponent to which 5 must be raised in order to obtain $7 .$ This is $\log _{5} 7$
(a) Use the change-of-base rule and your calculator to find $\log _{5} 7$
(b) Raise 5 to the number you found in part (a). What is your result?
(c) Using as many decimal places as your calculator gives, write the solution set of $5^{x}=7$.

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01:41

Problem 59

Let $m$ be the number of letters in your first name, and let $n$ be the number of letters in your last name.
(a) In your own words, explain what log, $n$ means.
(b) Use your calculator to find $\log _{m} n$
(c) Raise $m$ to the power indicated by the number found in part (b). What is your result?

Amy J.
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01:59

Problem 60

The value of $e$ can be expressed as
$$e=1+\frac{1}{1}+\frac{1}{1 \cdot 2}+\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{1 \cdot 2 \cdot 3 \cdot 4}+\cdots$$
Approximate $e$ using two terms of this expression, then three terms, four terms, five terms, and six terms. How close is the approximation to the value of $e \approx 2.718281828$ with six terms? Does this infinite sum approach the value of $e$ very quickly?

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01:13

Problem 61

For $1981-2003$, the number of billion cubic feet of natural gas gross withdrawals from crude oil wells in the United States can be approximated by the function defined by
$$f(x)=3800+585 \log _{2} x$$
where $x=1$ represents $1981, x=2$ represents $1982,$ and so on. (Source: Energy Information Administration.) Use this function to approximate the number of cubic feet withdrawn in $2003,$ to the nearest unit.

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01:21

Problem 62

According to selected from the last two decades of the 20 th century, the number of trillion cubic feet of dry natural gas consumed worldwide can be approximated by the function defined by
$$f(x)=51.47+6.044 \log _{2} x$$
where $x=1$ represents $1980, x=2$ represents $1981,$ and so on. (Source: Energy Information Administration.) Use this function to approximate consumption in $2003,$ to the nearest hundredth.

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00:31

Problem 63

Solve equation.
$4^{2 x}=8^{3 x+1}$

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00:41

Problem 64

Solve equation.
$2^{5 x}=\left(\frac{1}{16}\right)^{x+3}$

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00:18

Problem 65

Solve equation.
$\log _{3}(x+4)=2$

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00:27

Problem 66

Solve equation.
$\log _{x} 64=2$

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00:26

Problem 67

Solve equation.
$\log _{1 / 2} 8=x$

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00:22

Problem 68

Solve equation.
$\log _{a} 1=0$

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00:26

Problem 69

Write as a single logarithm. Assume $x>0$.
$\log (x+2)+\log (x+3)$

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00:36

Problem 70

Write as a single logarithm. Assume $x>0$.
$\log _{4}(x+4)-2 \log _{4}(3 x+1)$

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