Chapter 4

Describe the product rule for logarithms and give an example.

Evaluate each expression without using a calculator. $$ \log _{5}\left(\log _{2} 32\right) $$

Describe the quotient rule for logarithms and give an example.

Evaluate each expression without using a calculator. $$ \log _{2}\left(\log _{3} 81\right) $$

Describe the power rule for logarithms and give an example.

Evaluate each expression without using a calculator. $$ \log (\ln e) $$

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

Find the domain of each logarithmic function. $$ f(x)=\ln \left(x^{2}-x-2\right) $$

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

Find the domain of each logarithmic function. $$ f(x)=\ln \left(x^{2}-4 x-12\right) $$

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

Find the domain of each logarithmic function. $$ f(x)=\log \left(\frac{x+1}{x-5}\right) $$

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

Find the domain of each logarithmic function. $$ f(x)=\log \left(\frac{x-2}{x+5}\right) $$

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

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. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(y=2+\log _{3} x, y=\log _{3}(x+2),\) and \(y=-\log _{3} x\) in the same vicwing 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.

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. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

The bar graph indicates that the percentage of first-year college students expressing antifeminist views declined after \(1970 .\) Use this information to solve. (GRAPH CANNOT COPY). The function $$f(x)=-7.52 \ln x+53$$ models the percentage of first-year college men, \(f(x)\) expressing antifeminist views (by agreeing with the statement) \(x\) years after 1969. a. Use the function to find the percentage of first-year college men expressing antifeminist views in 2008 . Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph? By how much? b. Use the function to project the percentage of first-year college men who will express antifeminist views in 2015 . Round to one decimal place.

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

The bar graph indicates that the percentage of first-year college students expressing antifeminist views declined after \(1970 .\) Use this information to solve. (GRAPH CANNOT COPY). The function $$f(x)=-4.82 \ln x+32.5$$ models the percentage of first-year college women, \(f(x)\) expressing antifeminist views (by agreeing with the statement) \(x\) years after 1969. a. Use the function to find the percentage of first-year college women expressing antifeminist views in 2008 . Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph? By how much? b. Use the function to project the percentage of first-year college women who will express antifeminist views in \(2015 .\) Round to one decimal place.

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. (Any exposure to sounds of 130 decibels or higher puts a person at immediate risk for hearing damage.) Use the formula to solve. 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?

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. (Any exposure to sounds of 130 decibels or higher puts a person at immediate risk for hearing damage.) Use the formula to solve. What is the decibel level of a normal conversation, \(3.2 \times 10^{-6}\) watt per meter \(^{2} ?\)

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.

The \(p H\) scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a \(p H\) greater than \(7 .\) The lower the \(p H\) below \(7,\) the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. (Graph can't copy) The \(p H\) of a solution is given by $$ \mathrm{pH}=-\log x $$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. Normal, unpolluted rain has a \(\mathrm{pH}\) of about \(5.6 .\) What is the hydrogen ion concentration? b. An environmental concern involves the destructive effects of acid rain. The most acidic rainfall ever had a \(\mathrm{pH}\) of \(2.4 .\) What was the hydrogen ion concentration? c. How many times greater is the hydrogen ion concentration of the acidic rainfall in part (b) than the normal rainfall in part (a)?

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

The \(p H\) scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a \(p H\) greater than \(7 .\) The lower the \(p H\) below \(7,\) the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. (Graph can't copy) The \(p H\) of a solution is given by $$ \mathrm{pH}=-\log x $$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. The figure indicates that lemon juice has a pH of 2.3. What is the hydrogen ion concentration? b. Stomach acid has a pH that ranges from 1 to 3. What is the hydrogen ion concentration of the most acidic stomach? c. How many times greater is the hydrogen ion concentration of the acidic stomach in part (b) than the lemon juice in part (a)?

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

In Exercises \(121-124\), determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I cannot simplify the expression \(b^{m}+b^{n}\) by adding exponents, there is no property for the logarithm of a sum.

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

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

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

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

Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\)

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

In Exercises \(121-124\), determine whether each statement makes sense or does not make sense, and explain your reasoning. I expanded \(\log _{4} \sqrt{\frac{x}{y}}\) by writing the radical using a rational exponent and then applying the quotient rule, obtaining \(\frac{1}{2} \log _{4} x-\log _{4} y\)

In many states, a \(17 \%\) risk of a car accident with a blood alcohol concentration of 0.08 is the lowest level for charging a motorist with driving under the influence. Do you agree with the \(17 \%\) risk as a cutoff percentage, or do you feel that the percentage should be lower or higher? Explain your answer. What blood alcohol concentration corresponds to what you believe is an appropriate percentage?

Explain how to find the domain of a logarithmic function.

In Exercises \(125-128,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \ln \sqrt{2}=\frac{\ln 2}{2} $$

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. $$ 2^{x+1}=8 $$

Logarithmic models are well suited to phenomena in which growth is initially rapid but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.

In Exercises \(125-128,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \frac{\log _{7} 49}{\log _{7} 7}=\log _{7} 49-\log _{7} 7 $$

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 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. $$ \log _{3}(4 x-7)=2 $$

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$ f(x)=\ln x, g(x)=\ln (x+3) $$

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. $$ \log _{3}(3 x-2)=2 $$

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$ f(x)=\ln x, g(x)=\ln x+3 $$

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. $$ \log (x+3)+\log x=1 $$

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$ f(x)=\log x, g(x)=-\log x $$