Chapter 4

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

Evaluate or simplify each expression without using a calculator. $$e^{\ln 300}$$

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

Evaluate or simplify each expression without using a calculator. $$\ln e^{9 x}$$

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. \(\ln (5 x)+\ln 1=\ln (5 x)\)

Solve each equation. $$2|\ln x|-6=0$$

Evaluate or simplify each expression without using a calculator. $$\ln e^{13 x}$$

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

Solve each equation. $$3|\log x|-6=0$$

Evaluate or simplify each expression without using a calculator. $$e^{\ln 5 x^{2}}$$

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. \(\log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)}\)

Evaluate or simplify each expression without using a calculator. $$e^{\ln 7 x^{2}}$$

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

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

Write each equation in its equivalent exponential form. Then solve for x. $$\log _{3}(x-1)=2$$

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

Write each equation in its equivalent exponential form. Then solve for x. $$\log _{5}(x+4)=2$$

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

Write each equation in its equivalent exponential form. Then solve for x. $$\log _{4} x=-3$$

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?

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?

Write each equation in its equivalent exponential form. Then solve for x. $$\log _{64} x=\frac{2}{3}$$

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 \(0.03 ?\)

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?

Evaluate each expression without using a calculator. $$\log _{3}\left(\log _{7} 7\right)$$

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.

Complete the table for a savings account subject to \(n\) compoundings yearly \(\left[A=P\left(1+\frac{r}{n}\right)^{m}\right]\). Round answers to one decimal place. Amount Invested 7250 dollar Number of Compounding Periods 12 Annual Interest Rate 6.5% A ccumulated Amount 15,000 dollar Time \(t\) in Years ________

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 \(\log _{14} 283\).

Complete the table for a savings account subject to contimuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. Amount Invested 8000 dollar Annual Interest Rate 8% Accumulated Amount Double the amount invested Time \(t\) in Years _______

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.

Complete the table for a savings account subject to contimuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. Amount Invested 8000 dollar Annual Interest Rate 20.3% Accumulated Amount 12,000 dollar Time \(t\) in Years _______

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.

Complete the table for a savings account subject to contimuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. Amount Invested 2350 dollar Annual Interest Rate 15.7% Accumulated Amount Triple the amount invested Time \(t\) in Years _______

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 \(13 ?\)

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 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.

Complete the table for a savings account subject to contimuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. Amount Invested 17,425 dollar Annual Interest Rate 4.25% Accumulated Amount 25,000 dollar Time \(t\) in Years _______