Chapter 4

In Exercises \(75-82,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$3^{x+1}=9$$

Use inverse properties of logarithms to simplify each expression. $$\ln e^{13 x}$$

In Exercises \(75-82,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log _{3}(4 x-7)=2$$

Use inverse properties of logarithms to simplify each expression. $$e^{\ln 5 x^{2}}$$

In Exercises \(75-82,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log _{3}(3 x-2)=2$$

Use inverse properties of logarithms to simplify each expression. $$e^{\ln 7 x^{2}}$$

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function. $$ y=\log _{3} x $$

In Exercises \(75-82,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log (x+3)+\log x=1$$

Use inverse properties of logarithms to simplify each expression. $$10^{\log \sqrt{x}}$$

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function.$$ y=\log _{15} x $$

In Exercises \(75-82,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log (x-15)+\log x=2$$

Use inverse properties of logarithms to simplify each expression. $$10^{\log \sqrt[3]{x}}$$

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function. $$ y=\log _{2}(x+2) $$

In Exercises \(75-82,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$3^{x}=2 x+3$$

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the function to solve Exercises. Approximately what percent of her adult height is a girl at age \(13 ?\)

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function. $$ y=\log _{3}(x-2) $$

In Exercises \(75-82,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$5^{x}=3 x+4$$

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the function to solve Exercises. Approximately what percent of her adult height is a girl at age ten?

The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human car. The formula $$ D=10\left(\log I-\log I_{0}\right) $$ describes the loudness level of a sound, \(D\), in decibels, where \(I\) is the intensity of the sound, in watts per meter". and \(I_{0}\) is the intensity of a sound barely audible to the human ear. a. Express the formula so that the expression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at \(a\) distance of \(x\) miles from the eye of a hurricane. Use this function to solve Exercises \(83-84\) Graph the function in a \([0,500,50]\) by \([27,30,1]\) viewing rectangle. What does the shape of the graph indicate about barometric air pressure as the distance from the eye increases?

The annual amount that we spend to attend sporting events can be modeled by $$f(x)=2.05+1.3 \ln x$$ where \(x\) represents the number of years after 1984 and \(f(x)\) represents the total annual expenditures for admission to spectator sports, in billions of dollars. In \(2000,\) approximately how much was spent on admission to spectator sports?

The formula $$ t=\frac{1}{c}[\ln A-\ln (A-N)] $$ describes the time, \(t,\) in weeks, that it takes to achieve mastery of a portion of a task, where \(A\) is the maximum learning possible, \(N\) is the portion of the learning that is to be achieved, and \(c\) is a constant used to measure an individual's learning style. a. Express the formula so that the expression in brackets is written as a single logarithm. b. The formula is also used to determine how long it will take chimpanzees and apes to master a task. For example, a typical chimpanzce learning sign language can master a maximum of 65 signs. Use the form of the formula from part (a) to answer this question: How many weeks will it take a chimpanzee to master 30 signs if \(c\) for that chimp is \(0.03 ?\)

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at \(a\) distance of \(x\) miles from the eye of a hurricane. Use this function to solve Exercises \(83-84\) Use an equation to answer this question: How far from the eye of a hurricane is the barometric air pressure 29 inches of mercury? Use the \([\mathrm{TRACE}]\) and \([\mathrm{ZOOM}]\) features or the intersect command of your graphing utility to verify your answer.

The percentage of U.S. households with cable television can be modeled by $$f(x)=18.32+15.94 \ln x$$ where \(x\) represents the number of years after 1979 and \(f(x)\) represents the percentage of U.S. households with cable television. What percentage of U.S. households had cable television in \(1990 ?\)

Describe the product rule for logarithms and give an example.

The function \(P(t)=145 e^{-0.022 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. \([\mathrm{TRACE}]\) along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where I is the intensity of the sound, in watts per meter.\(^{2}\) Decibel levels range from \(0,\) a barely audible sound, to \(160, a\) sound resulting in a ruptured eardrum. Use the formula to solve Exercises. The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter \(^{2}\). Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?

Describe the quotient rule for logarithms and give an example.

The function \(W(t)=2600\left(1-0.51 e^{-0.075 t}\right)^{3}\) models the weight, \(W(t),\) in kilograms, of a female African elephant at age \(t\) years. (1 kilogram \(\approx 2.2\) pounds) Use a graphing utility to graph the function. Then \([\text { TRACE }]\) along the curve to estimate the age of an adult female elephant weighing 1800 kilograms.

The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where I is the intensity of the sound, in watts per meter.\(^{2}\) Decibel levels range from \(0,\) a barely audible sound, to \(160, a\) sound resulting in a ruptured eardrum. Use the formula to solve Exercises. What is the decibel level of a normal conversation, \(3.2 \times 10^{-6}\) watt per meter \(^{2} ?\)

Describe the power rule for logarithms and give an example.

Which one of the following is true? a. If \(\log (x+3)=2,\) then \(e^{2}=x+3\) b. If \(\log (7 x+3)-\log (2 x+5)=4,\) then in exponential form \(10^{4}=(7 x+3)-(2 x+5)\) c. If \(x=\frac{1}{k} \ln y,\) then \(y=e^{k x}\) d. Examples of exponential equations include \(10^{x}=5.71, e^{x}=0.72,\) and \(x^{10}=5.71\)

Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, \(f(t),\) after \(t\) months was modeled by the function $$f(t)=88-15 \ln (t+1), \quad 0 \leq t \leq 12$$ a. What was the average score on the original exam? b. What was the average score after 2 months? 4 months? 6 months? 8 months? 10 months? one year? c. Sketch the graph of \(f\) (either by hand or with a graphing utility). Describe what the graph indicates in terms of the material retained by the students.

Without showing the details, explain how to condense \(\ln x-2 \ln (x+1)\)

If \(\ S 4000\) is deposited into an account paying \(3 \%\) interest compounded annually and at the same time \(\ S 2000\) is deposited into an account paying \(5 \%\) interest compounded annually, after how long will the two accounts have the same balance?

Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.

Describe the change-of-base property and give an example.

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\ln x)^{2}=\ln x^{2}$$

What question can be asked to help evaluate \(\log _{3} 81 ?\)

Explain how to use your calculator to find \(\log _{14} 283\).

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$

Explain why the logarithm of 1 with base \(b\) is 0.

You overhear a student talking about a property of logarithms in which division becomes subtraction. Explain what the student means by this.

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

Describe the following property using words: \(\log _{h} b^{x}=x.\)

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; 1-\frac{1}{2}+\frac{1}{3} ; 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5} ; \ldots .\) Describe what you observe.

Rescarch applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: \(\mathrm{pH}\) (acidity of solutions), intensity of sound (decibels), brightness of stars, consumption of natural resources, human memory, progress over time in a sport, profit over time. For the area that you select. explain how logarithmic functions are used and provide examples.

Explain how to use the graph of \(f(x)=2^{x}\) to obtain the graph of \(g(x)=\log _{2} x.\)

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(\quad y=2+\log _{3} x, \quad y=\log _{3}(x+2), \quad\) and \(y=-\log _{3} x\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.

Explain how to find the domain of a logarithmic function.