Chapter 4

Solve each equation. $$ 5^{x^{2}}=50 $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ e^{\ln 7 x^{-1}} $$

will help you prepare for the material covered in the next section. 25 to what power gives \(5 ?\left(25^{\prime}=5\right)\)

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. $$ \log _{6}\left(\frac{x-1}{x^{2}+4}\right)=\log _{6}(x-1)-\log _{6}\left(x^{2}+4\right) $$

Solve each equation. $$ \ln (2 x+1)+\ln (x-3)-2 \ln x=0 $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ 10^{\log \sqrt{x}} $$

will help you prepare for the material covered in the next section. Solve: \((x-3)^{2}>0\)

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. $$ \log _{6}[4(x+1)]=\log _{6} 4+\log _{6}(x+1) $$

Solve each equation. $$ \ln 3-\ln (x+5)-\ln x=0 $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ 10^{\log \sqrt{x}} $$

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. $$ \log _{3} 7=\frac{1}{\log _{7} 3} $$

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

In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. $$ \log _{3}(x-1)=2 $$

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

In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. $$ \log _{5}(x+4)=2 $$

The formula \(A=37.3 e^{0.0095 t}\) models the population of California, \(A,\) in millions, \(t\) years after 2010 . a. What was the population of California in \(2010 ?\) b. When will the population of California reach 40 million?

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?

In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. $$ \log _{4} x=-3 $$

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 \(0 . \overline{03 ?}\)

The formula \(A=25.1 e^{0.0187 t}\) models the population of Texas, \(A,\) in millions, \(t\) years after 2010 . a. What was the population of Texas in \(2010 ?\) b. When will the population of Texas reach 28 million?

In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. $$ \log _{64} x=\frac{2}{3} $$

Describe the product rule for logarithms and give an example.

In Exercises 105–108, evaluate each expression without using a calculator. $$ \log _{3}\left(\log _{7} 7\right) $$

Describe the quotient rule for logarithms and give an example.

In Exercises 105–108, evaluate each expression without using a calculator. $$ \log _{5}\left(\log _{2} 32\right) $$

Describe the power rule for logarithms and give an example.

In Exercises 105–108, evaluate each expression without using a calculator. $$ \log _{2}\left(\log _{3} 81\right) $$

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

In Exercises 105–108, evaluate each expression without using a calculator. $$ \log (\ln e) $$

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

In Exercises 109–112, find the domain of each logarithmic function. $$ f(x)=\ln \left(x^{2}-x-2\right) $$

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

In Exercises 109–112, find the domain of each logarithmic function. $$ f(x)=\ln \left(x^{2}-4 x-12\right) $$

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

In Exercises 109–112, find the domain of each logarithmic function. $$ f(x)=\log \left(\frac{x+1}{x-5}\right) $$

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.

In Exercises 109–112, find the domain of each logarithmic function. $$ f(x)=\log \left(\frac{x-2}{x+5}\right) $$

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 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.

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 113–114. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

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 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 113–114. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?

By 2019 , nearly \(\$$ I out of every \)\$ 5\( spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product \)(G D P)\( going toward health care from 2007 through \)2014,\( with a projection for 2019 The data can be modeled by the function \)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 .\( Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \)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 \(b>1\)

By 2019 , nearly \(\$$ I out of every \)\$ 5\( spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product \)(G D P)\( going toward health care from 2007 through \)2014,\( with a projection for 2019 The data can be modeled by the function \)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 .\( Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \)2008 .\( Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \)18.6 \%$ of the U.S. gross domestic product go toward health care? Round to the nearest year.

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 meter2. 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 Exercises 117–118. 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 function \(P(x)=95-30 \log _{2} x\) models the percentage, \(P(x),\) of students who could recall the important features of a classroom lecture as a function of time, where \(x\) represents the number of days that have elapsed since the lecture was given. The figure at the top of the next column shows the graph of the function. Use this information to solve Exercises \(117-118\). Round answers to one decimal place. After how many days have all students forgotten the important features of the classroom lecture? (Let \(P(x)=0\) and solve for \(x\).) Locate the point on the graph that conveys this information.

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 meter2. 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 Exercises 117–118. What is the decibel level of a normal conversation, \(3.2 \times 10^{-6}\) watt per meter \(^{2} ?\)

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 meter2. 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 Exercises 117–118. 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.

Explaining the Concepts. 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.