Chapter 3

Write the expression as the logarithm of a single quantity. $$\ln 3+\frac{1}{2} \ln x+\ln y-\frac{1}{3} \ln z$$

Solve the equation for \(x\). $$8^{x}=\left(\frac{1}{32}\right)^{x-2}$$

During a flu epidemic, the number of children in the Woodbridge Community School System who contracted influenza after \(t\) days was given by $$ Q(t)=\frac{1000}{1+199 e^{-0.8 t}} $$ a. How many children were stricken by the flu after the first day? b. How many children had the flu after 10 days?

Solve the equation for \(x\). $$3^{x-x^{2}}=\frac{1}{9^{x}}$$

The change from religious to lay teachers at Roman Catholic schools has been partly attributed to the decline in the number of women and men entering religious orders. The percentage of teachers who are lay teachers is given by $$ f(t)=\frac{98}{1+2.77 e^{-t}} \quad(0 \leq t \leq 4) $$ where \(t\) is measured in decades, with \(t=0\) corresponding to the beginning of 1960 . What percentage of teachers were lay teachers at the beginning of 1990 ?

Solve the equation for \(x\). $$3^{2 x}-12 \cdot 3^{x}+27=0$$

On the basis of data collected during an experiment, a biologist found that the growth of a fruit fly (Drosophila) with a limited food supply could be approximated by the exponential model $$ N(t)=\frac{400}{1+39 e^{-0.16 t}} $$ where \(t\) denotes the number of days since the beginning of the experiment. a. What was the initial fruit fly population in the experiment? b. What was the population of the fruit fly colony on the 20th day?

Use the laws of logarithms to expand and simplify the expression. $$\log x\left(x^{2}+1\right)^{-1 / 2}$$

Solve the equation for \(x\). $$2^{2 x}-4 \cdot 2^{x}+4=0$$

The number of citizens aged \(45-64 \mathrm{yr}\) is projected to be $$ P(t)=\frac{197.9}{1+3.274 e^{-0.0361 t}} \quad(0 \leq t \leq 20) $$ where \(P(t)\) is measured in millions and \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1990\. People belonging to this age group are the targets of insurance companies that want to sell them annuities. What is the projected population of citizens aged \(45-64 \mathrm{yr}\) in \(2010 ?\)

Use the laws of logarithms to expand and simplify the expression. $$\log \frac{\sqrt{x+1}}{x^{2}+1}$$

Sketch the graphs of the given functions on the same axes. \(y=2^{x}, y=3^{x}\), and \(y=4^{x}\)

The U.S. population is approximated by the function $$ P(t)=\frac{616.5}{1+4.02 e^{-0.5 t}} $$ where \(P(t)\) is measured in millions of people and \(t\) is measured in 30 -yr intervals, with \(t=0\) corresponding to 1930 . What is the expected population of the United States in \(2020(t=3) ?\)

Use the laws of logarithms to expand and simplify the expression. $$\ln \frac{e^{x}}{1+e^{x}}$$

Sketch the graphs of the given functions on the same axes. \(y=\left(\frac{1}{2}\right)^{x}, y=\left(\frac{1}{3}\right)^{x}\), and \(y=\left(\frac{1}{4}\right)^{x}\)

Three hundred students attended the dedication ceremony of a new building on a college campus. The president of the traditionally female college announced a new expansion program, which included plans to make the college coeducational. The number of students who learned of the new program \(t \mathrm{hr}\) later is given by the function $$ f(t)=\frac{3000}{1+B e^{-k t}} $$ If 600 students on campus had heard about the new program \(2 \mathrm{hr}\) after the ceremony, how many students had heard about the policy after \(4 \mathrm{hr}\) ?

Sketch the graphs of the given functions on the same axes. \(y=2^{-x}, y=3^{-x}\), and \(y=4^{-x}\)

A radioactive substance decays according to the formula $$ Q(t)=Q_{0} e^{-k t} $$ where \(Q(t)\) denotes the amount of the substance present at time \(t\) (measured in years), \(Q_{0}\) denotes the amount of the substance present initially, and \(k\) (a positive constant) is the decay constant. a. Show that half-life of the substance is \(\bar{t}=\ln 2 / k\). b. Suppose a radioactive substance decays according to the formula $$ Q(t)=20 e^{-0.000123 \mathrm{x} i} $$ How long will it take for the substance to decay to half the original amount?

Use the laws of logarithms to expand and simplify the expression. $$\ln x(x+1)(x+2)$$

Sketch the graphs of the given functions on the same axes. \(y=4^{0.5 x}\) and \(y=4^{-0.5 x}\)

Consider the logistic growth function $$ Q(t)=\frac{A}{1+B e^{-k r}} $$ Suppose the population is \(Q_{1}\) when \(t=t_{1}\) and \(Q_{2}\) when \(t=t_{2}\). Show that the value of \(k\) is $$ k=\frac{1}{t_{2}-t_{1}} \ln \left[\frac{Q_{2}\left(A-Q_{1}\right)}{Q_{1}\left(A-Q_{2}\right)}\right] $$

Use the laws of logarithms to expand and simplify the expression. $$\ln \frac{x^{1 / 2}}{x^{2} \sqrt{1+x^{2}}}$$

Sketch the graphs of the given functions on the same axes. \(y=4^{0.5 x}, y=4^{x}\), and \(y=4^{2 x}\)

Use the laws of logarithms to expand and simplify the expression. $$\ln \frac{x^{2}}{\sqrt{x}(1+x)^{2}}$$

Sketch the graphs of the given functions on the same axes. \(y=e^{x}, y=2 e^{x}\), and \(y=3 e^{x}\)

Sketch the graphs of the given functions on the same axes. \(y=e^{0.5 x}, y=e^{x}\), and \(y=e^{1.5 x}\)

Use the laws of logarithms to solve the equation. $$\log _{3} x=2$$

Sketch the graphs of the given functions on the same axes. \(y=e^{-0.5 x}, y=e^{-x}\), and \(y=e^{-1.5 x}\)

Use the laws of logarithms to solve the equation. $$\log _{2} 8=x$$

Sketch the graphs of the given functions on the same axes. \(y=0.5 e^{-x}, y=e^{-x}\), and \(y=2 e^{-x}\)

Use the laws of logarithms to solve the equation. $$\log _{3} 27=2 x$$

Sketch the graphs of the given functions on the same axes. \(y=1-e^{-x}\) and \(y=1-e^{-0.5 x}\)

Use the laws of logarithms to solve the equation. $$\log _{x} 10^{3}=3$$

A function \(f\) has the form \(f(x)=A e^{k x}\). Find \(f\) if it is known that \(f(0)=100\) and \(f(1)=120\). Hint: \(e^{k t}=\left(e^{k}\right)^{x}\).

Use the laws of logarithms to solve the equation. $$\log _{x} \frac{1}{16}=-2$$

Use the laws of logarithms to solve the equation. $$\log _{2}(2 x+5)=3$$

If $$ f(t)=\frac{1000}{1+B e^{-b t}} $$ find \(f(5)\) given that \(f(0)=20\) and \(f(2)=30\). Hint: \(e^{k n}=\left(e^{k}\right)^{x}\)

Use the laws of logarithms to solve the equation. $$\log _{4}(5 x-4)=2$$

Employers are increasingly turning to GPS (global positioning system) technology to keep track of their fleet vehicles. The estimated number of automatic vehicle trackers installed on fleet vehicles in the United States is approximated by $$ N(t)=0.6 e^{0.17 t} \quad(0 \leq t \leq 5) $$ where \(N(t)\) is measured in millions and \(t\) is measured in years, with \(t=0\) corresponding to 2000 . a. What was the number of automatic vehicle trackers installed in the year \(2000 ?\) How many were projected to be installed in \(2005 ?\) b. Sketch the graph of \(N\).

Use the laws of logarithms to solve the equation. $$\log _{2} x-\log _{2}(x-2)=3$$

Because of medical technology advances, the disability rates for people over \(65 \mathrm{yr}\) old have been dropping rather dramatically. The function $$ R(t)=26.3 e^{-0.016} \quad(0 \leq t \leq 18) $$ gives the disability rate \(R(t)\), in percent, for people over age 65 from \(1982(t=0)\) through 2000 , where \(t\) is measured in years. a. What was the disability rate in \(1982 ?\) In \(1986 ?\) In 1994 ? In 2000 ? b. Sketch the graph of \(R\).

Use the laws of logarithms to solve the equation. $$\log x-\log (x+6)=-1$$

The percentage of families that were married households between 1970 and 2000 is approximately $$ P(t)=86.9 e^{-0.05 t} \quad(0 \leq t \leq 3) $$ where \(t\) is measured in decades, with \(t=0\) corresponding to the beginning of 1970 . a. What percentage of families were married households at the beginning of \(1970,1980,1990\), and 2000 ? b. Sketch the graph of \(P\).

Use the laws of logarithms to solve the equation. $$\log _{5}(2 x+1)-\log _{5}(x-2)=1$$

According to a study conducted in 2000 , the projected number of Web addresses (in billions) is approximated by the function $$ N(t)=0.45 e^{0.5696} \quad(0 \leq t \leq 5) $$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1997 . a. Complete the following table by finding the number of Web addresses in each year: b. Sketch the graph of \(N\).

Use the laws of logarithms to solve the equation. $$\log (x+7)-\log (x-2)=1$$

The number of Internet users in China is projected to be $$ N(t)=94.5 e^{0.2 t} \quad(1 \leq t \leq 6) $$ where \(N(t)\) is measured in millions and \(t\) is measured in years, with \(t=1\) corresponding to the beginning of 2005 . a. How many Internet users were there at the beginning of \(2005 ?\) At the beginning of 2006 ? b. How many Internet users are there expected to be at the beginning of 2010 ? c. Sketch the graph of \(N\).

The alternative minimum tax was created in 1969 to prevent the very wealthy from using creative deductions and shelters to avoid having to pay anything to the Internal Revenue Service. But it has increasingly hit the middle class. The number of taxpayers subjected to an alternative minimum tax is projected to be $$ N(t)=\frac{35.5}{1+6.89 e^{-0.8674 t}} \quad(0 \leq t \leq 6) $$ where \(N(t)\) is measured in millions and \(t\) is measured in years, with \(t=0\) corresponding to 2004 . What is the projected number of taxpayers subjected to an alternative minimum tax in 2010 ?

Use the laws of logarithms to solve the equation. $$\log _{3}(x+1)+\log _{3}(2 x-3)=1$$

The concentration of a drug in an organ at any time \(t\) (in seconds) is given by $$ C(t)=\left\\{\begin{array}{ll} 0.3 t-18\left(1-e^{-260}\right) & \text { if } 0 \leq t \leq 20 \\ 18 e^{-560}-12 e^{-(t-20) \sqrt{6}} & \text { if } t>20 \end{array}\right. $$ where \(C(t)\) is measured in grams/cubic centimeter \(\left(\mathrm{g} / \mathrm{cm}^{3}\right)\). a. What is the initial concentration of the drug in the organ? b. What is the concentration of the drug in the organ after 10 sec? c. What is the concentration of the drug in the organ after 30 sec?