Chapter 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. 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 data can be modeled by the function \(f(x)=1.2 \ln x+15.7\) where \(f(x)\) is the percentage of the U.S. gross domestic product going toward health care \(x\) years after \(2006 .\) a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \(2009 .\) Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \(18.5 \%\) of the U.S. gross domestic product go toward health care? Round to the nearest year.

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 fi rst-year college students expressing antifeminist views declined after 1970. CAN'T COPY THE GRAPH 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 I3 0 decibels or higher puts a person at immediate risk for hearing damage.) The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter? 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 I3 0 decibels or higher puts a person at immediate risk for hearing damage.) What is the decibel level of a normal conversation, \(3.2 \times 10^{-6}\) watt per meter \(^{2} ?\)

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. 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. Express answers as powers of $10. a. Normal, unpolluted rain has a 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 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)? (pH SCALE CAN'T COPY)

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.

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

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

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

Determine whether 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 cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

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

Determine whether statement makes sense or does not make sense, and explain your reasoning. Because logarithms are exponents, the product, quotient, and power rules remind me of properties for operations with exponents.

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

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

Determine whether statement makes sense or does not make sense, and explain your reasoning. I can use any positive number other than 1 in the change-of-base property, but the only practical bases are 10 and \(e\) because my calculator gives logarithms for these two bases.

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

Determine whether 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 \(\frac{1}{2} \log _{4} x-\log _{4} y\).

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

Explain how to find the domain of a logarithmic function.

Determine whether 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. $$3^{x+1}=9$$

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.

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

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

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 \(f\). $$f(x)=\ln x, g(x)=\ln x+3$$

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. $$\log (x-15)+\log x=2$$

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

If \(\log 3=A\) and \(\log 7=B,\) find \(\log _{7} 9\) in terms of \(A\) and \(B\).

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. $$3^{x}=2 x+3$$

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\log x, g(x)=\log (x-2)+1$$

Write as a single term that does not contain a logarithm: $$e^{\ln 8 x^{5}-\ln 2 x^{2}}$$

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. $$5^{x}=3 x+4$$

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1), \quad\) where \(\quad 0 \leq t \leq 12 . \quad\) Use \(\quad\) a graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65.

If \(f(x)=\log _{b} x,\) show that $$\frac{f(x+h)-f(x)}{h}=\log _{b}\left(1+\frac{h}{x}\right)^{\frac{1}{h}}, h \neq 0.$$

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. 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?

In parts (a)-(c), graph \(f\) and \(g\) in the same viewing rectangle. a. \(f(x)=\ln (3 x), g(x)=\ln 3+\ln x\) b. \(f(x)=\log \left(5 x^{2}\right), g(x)=\log 5+\log x^{2}\) c. \(f(x)=\ln \left(2 x^{3}\right), g(x)=\ln 2+\ln x^{3}\) d. Describe what you observe in parts (a)-(c). Generalize this observation by writing an equivalent expression for \(\log _{b}(M N),\) where \(M>0\) and \(N>0\) e. Complete this statement: The logarithm of a product is equal to _________.

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 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.

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. \(y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}\)

Will help you prepare for the material covered in the next section. Solve for \(x: a(x-2)=b(2 x+3)\)

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. The function \(P(t)=145 e^{-0.092 t}\) 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. 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.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote.

Will help you prepare for the material covered in the next section. Solve: \(x(x-7)=3\)