Chapter 4

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.

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

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.

Determine whether each 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.

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

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

Determine whether each 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.

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

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?

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

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

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

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

Explaining the Concepts. Explain how to find the domain of a logarithmic function.

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

Explaining the Concepts. 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.

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

In Exercises \(128-131\), 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. $$ \log (x+3)+\log x=1 $$

In Exercises \(128-131\), 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. $$ \log (x-15)+\log x=2 $$

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

In Exercises \(128-131\), graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f\) $$ f(x)=\log x, g(x)=-\log x $$

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

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

In Exercises \(128-131,\) graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f\) $$ f(x)=\log x, g(x)=\log (x-2)+1 $$

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

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

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 \(\quad 0 \leq t \leq 12 .\) Use a graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65.

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. 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?

Use the proof of the product rule in the appendix to prove the quotient rule.

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. \(f(x)=\log \left(5 x^{2}\right), g(x)=\log 5+\log x^{2}\) \(f(x)=\ln \left(2 x^{3}\right), g(x)=\ln 2+\ln x^{3}\) 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 this function to solve. 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 TRACE and ZOOM features or the intersect command of your graphing utility to verify your answer.

Given \(f(x)=\frac{2}{x+1}\) and \(g(x)=\frac{1}{x},\) find each of the following: a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) (Section \(2.6, \text { Example } 6)\)

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. $$ y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2} $$

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.

Use the Leading Coefficient Test to determine the end behavior of the graph of \(f(x)=-2 x^{2}(x-3)^{2}(x+5)\) (Section \(3.2,\) Example 2 )

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

Graph: \(f(x)=\frac{4 x^{2}}{x^{2}-9}\) (Section 3.5, Example 6)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations $$\log (3 x+1)=5 \text { and } \log (3 x+1)=\log 5$$ are similar, I solved them using the same method.

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can solve \(4^{x}=15\) by writing the equation in logarithmic form.

Will help you prepare for the material covered in the next section. $$ \text { Solve: } \frac{x+2}{4 x+3}=\frac{1}{x} $$

In Exercises 139–142, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \frac{\log _{2} 8}{\log _{2} 4}=\frac{8}{4} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It's important for me to check that the proposed solution of an equation with logarithms gives only logarithms of positive numbers in the original equation.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \text { If } \log (x+3)=2, \text { then } e^{2}=x+3 $$

In Exercises 139–142, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\).