Chapter 9

The number of people \(N\) who will receive a forwarded e-mail can be approximated by \(N=\frac{P}{1+(P-S) e^{-0.35 t}},\) where \(P\) is the total number of people online, \(S\) is the number of people who start the e-mail, and \(t\) is the time in minutes. Suppose four people want to send an e-mail to all those who are online at that time. If there are 156,000 people online, how many people will have received the e-mail after 25 minutes?

Evaluate each expression. $$ \log _{3} 81 $$

For Exercises 55 and 56 , use the following information. If you deposit \(P\) dollars into a bank account paying an annual interest rate \(r\) (expressed as a decimal), with \(n\) interest payments each year, the amount \(A\) you would have after \(t\) years is \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) Marta places \(\$ 100\) in a savings account earning 2\(\%\) annual interest, compounded quarterly. If Marta adds no more money to the account, how long will it take the money in the account to reach \(\$ 125 ?\)

Solve each equation or inequality. Check your solutions. \(\log _{2}(3 x-8) \geq 6\)

The number of people \(N\) who will receive a forwarded e-mail can be approximated by \(N=\frac{P}{1+(P-S) e^{-0.35 t}},\) where \(P\) is the total number of people online, \(S\) is the number of people who start the e-mail, and \(t\) is the time in minutes. Suppose four people want to send an e-mail to all those who are online at that time. How much time will pass before half of the people will receive the e-mail?

Evaluate each expression. $$ \log _{9} \frac{1}{729} $$

For Exercises 55 and 56 , use the following information. If you deposit \(P\) dollars into a bank account paying an annual interest rate \(r\) (expressed as a decimal), with \(n\) interest payments each year, the amount \(A\) you would have after \(t\) years is \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) Marta places \(\$ 100\) in a savings account earning 2\(\%\) annual interest, compounded quarterly. How long will it take for Marta's money to double?

Solve each equation or inequality. Check your solutions. \(\log _{6}(2 x-3)=\log _{6}(x+2)\)

Graph each pair of functions on the same screen. Then compare the graphs, listing both similarities and differences in shape, asymptotes, domain, range, and \(y\) -intercepts. $$ y=2^{x} \quad y=2^{x}+3 $$

Solve each equation. Round to the nearest ten-thousandth. \(\ln x+\ln 3 x=12\)

Evaluate each expression. $$ \log _{7} 7^{2 x} $$

Solve \(\log _{\sqrt{a}} 3=\log _{a} x\) for \(x\) and explain each step.

Solve each equation or inequality. Check your solutions. \(\log _{7}\left(x^{2}+36\right)=\log _{7} 100\)

Graph each pair of functions on the same screen. Then compare the graphs, listing both similarities and differences in shape, asymptotes, domain, range, and \(y\) -intercepts. $$ y=3^{x} \quad y=3^{x+1} $$

Solve each equation. Round to the nearest ten-thousandth. \(\ln 4 x+\ln x=9\)

Solve each equation or inequality. Check your solutions. $$ 3^{5 n+3}=3^{33} $$

Write \(\frac{\log _{5} 9}{\log _{5} 3}\) as a single logarithm.

Graph each pair of functions on the same screen. Then compare the graphs, listing both similarities and differences in shape, asymptotes, domain, range, and \(y\) -intercepts. $$ y=\left(\frac{1}{5}\right)^{x} \quad y=\left(\frac{1}{5}\right)^{x-2} $$

Solve each equation. Round to the nearest ten-thousandth. \(\ln \left(x^{2}+12\right)=\ln x+\ln 8\)

Solve each equation or inequality. Check your solutions. $$ 7^{a}=49^{-4} $$

a. Find the values of \(\log _{2} 8\) and \(\log _{8} 2 .\) b. Find the values of \(\log _{9} 27\) and \(\log _{27} 9\) c. Make and prove a conjecture about the relationship between \(\log _{a} b\) and \(\log _{b} a .\)

Graph each pair of functions on the same screen. Then compare the graphs, listing both similarities and differences in shape, asymptotes, domain, range, and \(y\) -intercepts. $$ y=\left(\frac{1}{4}\right)^{x} \quad y=\left(\frac{1}{4}\right)^{x}-1 $$

Solve each equation. Round to the nearest ten-thousandth. \(\ln x+\ln (x+4)=\ln 5\)

Solve each equation or inequality. Check your solutions. $$ 3^{d+4}>9^{d} $$

Show that each statement is true. \(\log _{5} 25=2 \log _{5} 5\)

Describe the effect of changing the values of \(h\) and \(k\) in the equation \(y=2^{x-h}+k\)

Give an example of an exponential equation that requires using natural logarithms instead of common logarithms to solve.

PHYSICS If a stone is dropped from a cliff, the equation \(t=\frac{1}{4} \sqrt{d}\) represents the time \(t\) in seconds that it takes for the stone to reach the ground. If \(d\) represents the distance in feet that the stone falls, find how long it would take for a stone to hit the ground after falling from a 150 -foot cliff.

ACT/SAT If \(2^{4}=3^{x}\) , then what is the approximate value of \(x ?\) $$ \begin{array}{l}{\text { A } 0.63} \\ {\text { B } 2.34} \\ {\text { C } 2.52} \\ {\text { D } 4}\end{array} $$

Show that each statement is true. \(\log _{16} 2 \cdot \log _{2} 16=1\)

OPEN ENDED Give an example of a value of \(b\) for which \(y=b^{x}\) represents exponential decay.

Colby and Elsu are solving \(\ln 4 x=5 .\) Who is correct? Explain your reasoning. Colby \(\begin{aligned} \ln 4 x &=5 \\ 10^{\ln } 4 x &=10^{5} \\ 4 x &=100,000 \\\ x &=25,000 \end{aligned}\) Elsu \(\begin{aligned} \ln 4 x &=5 \\ e^{\ln 4 x} &=e^{5} \\ 4 x &=e^{5} \\ x &=\frac{e^{5}}{4} \\ & \times 37.1033 \end{aligned}\)

PREREQUISITE SKILL Solve each equation or inequality. Check your solutions. $$ \log _{3} x=\log _{3}(2 x-1) $$

REVIEW Which equation is equivalent to \(\log _{4} \frac{1}{16}=x ?\) $$ \begin{array}{l}{\mathbf{F} \frac{1^{4}}{16}=x^{4}} \\\ {\mathbf{G}\left(\frac{1}{16}\right)^{4}=x} \\ {\mathbf{H} \quad 4^{x}=\frac{1}{16}} \\ {\mathbf{J} \quad 4^{\frac{1}{16}}=x}\end{array} $$

Show that each statement is true. \(\log _{7}\left[\log _{3}\left(\log _{2} 8\right)\right]=0\)

REASONING Identify each function as linear, quadratic, or exponential. $$\begin{array}{llll}{\text { a. } y=3 x^{2}} & {\text { b. } y=4(3)^{x}} & {\text { c. } y=2 x+4} & {\text { d. } y=4(0.2)^{x}+1}\end{array}$$

Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. For all positive numbers \(x\) and \(y, \frac{\log x}{\log y}=\frac{\ln x}{\ln y}\)

PREREQUISITE SKILL Solve each equation or inequality. Check your solutions. $$ \log _{10} 2^{x}=\log _{10} 32 $$

Use \(\log _{7} 2 \approx 0.3562\) and \(\log _{7} 3 \approx 0.5646\) to approximate the value of each expression. $$ \log _{7} 16 $$

Sketch the graphs of \(y=\log _{\frac{1}{2}} x\) and \(y=\left(\frac{1}{2}\right)^{x}\) on the same axes. Then describe the relationship between the graphs.

CHALLENGE Decide whether the following statement is sometimes, always, or never true. Explain your reasoning. For a positive base b other than \(1, b^{x} > b^{y}\) if and only if \(x > y\)

PREREQUISITE SKILL Solve each equation or inequality. Check your solutions. $$ \log _{2} 3 x>\log _{2} 5 $$

Use \(\log _{7} 2 \approx 0.3562\) and \(\log _{7} 3 \approx 0.5646\) to approximate the value of each expression. $$ \log _{7} 27 $$

Sketch the graphs of \(y=\log _{3} x, y=\log _{3}(x+2), y=\log _{3} x-3 .\) Then describe the relationship between the graphs

A recent study showed that the number of Australian homes with a computer doubles every 8 months. Assuming that the number is increasing continuously, at approximately what monthly rate must the number of Australian computer owners be increasing for this to be true? A. 68% B. 8.66% C. 0.0866% D. 0.002%

PREREQUISITE SKILL Solve each equation or inequality. Check your solutions. $$ \log _{5}(4 x+3)<\log _{5} 11 $$

Use \(\log _{7} 2 \approx 0.3562\) and \(\log _{7} 3 \approx 0.5646\) to approximate the value of each expression. $$ \log _{7} 36 $$

The magnitude of an earthquake is measured on a logarithmic scale called the Richter scale. The magnitude \(M\) is given by \(M=\log _{10} x,\) where \(x\) represents the amplitude of the seismic wave causing ground motion. How many times as great is the amplitude caused by an earthquake with a Richter scale rating of 7 as an aftershock with a Richter scale rating of 4?

ACT/SAT If \(4^{x+2}=48,\) then \(4^{x}=\) $$ \begin{array}{l}{\text { A } 3.0} \\ {\text { B } 6.4} \\ {\text { C } 6.9} \\\ {\text { D } 12.0}\end{array} $$

Which is the first incorrect step in simplifying \(\log _{3} \frac{3}{48} ?\) \(\begin{array}{ll}{\text { Step } 1 :} & {\log _{3} \frac{3}{48}=\log _{3} 3-\log _{3} 48} \\ {\text { Step } 2 :} & {=1-16} \\ {\text { Step } 3 :} & {=-15}\end{array}\) F. Step 1 G. Step 2 H. Step 3 J. Each step is correct.