Chapter 12

Will help you prepare for the material covered in the next section. If \(f(x)=2 x-5,\) find \(f^{-1}(x)\)

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

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log x+\log (x+3)=\log 10$$

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\log _{3} 7=\frac{1}{\log _{7} 3}$$

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

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (x+3)+\log (x-2)=\log 14$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln (x-4)+\ln (x+1)=\ln (x-8)$$

a. Evaluate the expression in part (a) without using a calculator. b. Use your result from part (a) to write the expression in part (b) as a single logarithm whose coefficient is 1 a. \(\log _{3} 9\) b. \(\log _{3} x+4 \log _{3} y-2\)

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{2}(x-1)-\log _{2}(x+3)=\log _{2}\left(\frac{1}{x}\right)$$

a. Evaluate the expression in part (a) without using a calculator. b. Use your result from part (a) to write the expression in part (b) as a single logarithm whose coefficient is 1 a. \(\log _{2} 16\) b. \(\log _{2} x+5 \log _{2} y-4\)

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 nuptured 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? Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?

Solve each equation. $$5^{2 x} \cdot 5^{4 x}=125$$

a. Evaluate the expression in part (a) without using a calculator. b. Use your result from part (a) to write the expression in part (b) as a single logarithm whose coefficient is 1 a. \(\log _{25} 5\) b. \(\log _{25} x+\log _{25}\left(x^{2}-1\right)-\log _{25}(x+1)-\frac{1}{2}\)

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 nuptured eardrum. Use the formula to solve Exercises. What is the decibel level of a normal conversation, \(3.2 \times 10^{-6}\) watt per meter \(^{2} ?\)

Solve each equation. $$3^{x+2} \cdot 3^{x}=81$$

a. Evaluate the expression in part (a) without using a calculator. b. Use your result from part (a) to write the expression in part (b) as a single logarithm whose coefficient is 1 a. \(\log _{36} 6\) b. \(\log _{36} x+\log _{36}\left(x^{2}-4\right)-\log _{36}(x+2)-\frac{1}{2}\)

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, to the nearest tenth, 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 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 ear. 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 \(^{2}\), 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?

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

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 chimpanzee 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 \(0.03 ?\)

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

Solve each equation. $$\log _{2}(x-6)+\log _{2}(x-4)-\log _{2} x=2$$

Describe the product rule for logarithms and give an example.

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

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

Describe the quotient rule for logarithms and give an example.

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

Solve each equation. $$5^{x^{2}-12}=25^{2 x}$$

Describe the power rule for logarithms and give an example.

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

Solve each equation. $$3^{x^{2}-12}=9^{2 x}$$

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

Explain how to find the domain of a logarithmic function.

The formula \(A=36.1 e^{0.0126 t}\) models the population of California, \(A,\) in millions, \(t\) years after 2005 a. What was the population of California in \(2005 ?\) b. When will the population of California reach 40 million?

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

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.

The formula \(A=22.9 e^{0.0183 t}\) models the population of Texas, \(A,\) in millions, \(t\) years after 2005 a. What was the population of Texas in \(2005 ?\) b. When will the population of Texas reach 27 million?

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

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

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

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.

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

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 \(y=-\log _{3} x \quad\) 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.

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

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?

Complete the table for a savings account subject to n compounding periods per year \(\left[A=P\left(1+\frac{r}{n}\right)^{n t}\right]\) Round answers to one decimal place. $$\begin{array}{l|c|l|l|l} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Number of } \\ \text { Compounding } \\ \text { Periods } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 1000 & 360 & & \$ 1400 & 2 \\ \hline \end{array}$$

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 \(b>1\)

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 is modeled by the human memory function \(f(t)=75-10 \log (t+1),\) where \(0 \leq t \leq 12\) Use a graphing utility to graph the function. Then determine how many months will elapse before the average score falls below 65

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 _______.

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 8000 & & \$ 12,000 & 2 \\ \hline \end{array}$$