Chapter 4

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{6} 17 $$

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. $$ \log _{4}(x+2)-\log _{4}(x-1)=1 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used an exponential model with a positive growth rate to describe the depreciation in my car's value over four years.

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{14} 87.5 $$

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. $$ 2 \log _{3}(x+4)=\log _{3} 9+2 $$

Where necessary, round answers to the nearest percent. In college, we study large volumes of information information that, unfortunately, we do not often retain for very long. The function $$ f(x)=80 e^{-0.5 x}+20 $$ describes the percentage of information, \(f(x)\), that a particular person remembers \(x\) weeks after learning the information. a. Substitute 0 for \(x\) and, without using a calculator, find the percentage of information remembered at the moment it is first learned. b. Substitute 1 for \(x\) and find the percentage of information that is remembered after 1 week. c. Find the percentage of information that is remembered after 4 weeks. d. Find the percentage of information that is remembered after one year ( 52 weeks).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After 100 years, a population whose growth rate is \(3 \%\) will have three times as many people as a population whose growth rate is \(1 \%\)

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{16} 57.2 $$

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. $$ 3 \log _{2}(x-1)=5-\log _{2} 4 $$

Where necessary, round answers to the nearest percent. In \(1626,\) Peter Minuit convinced the Wappinger Indians to sell him Manhattan Island for \(\$ 24 .\) If the Native Americans had put the \(\$ 24\) into a bank account paying \(5 \%\) interest, how much would the investment have been worth in the year 2010 if interest were compounded a. monthly? b. continuously?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative.

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{0.1} 17 $$

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

In Exercises 75–80, find the domain of each logarithmic function. $$ f(x)=\log _{5}(x+4) $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because carbon- 14 decays exponentially, carbon dating can determine the ages of ancient fossils.

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{0.3} 19 $$

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

In Exercises 75–80, find the domain of each logarithmic function. $$ f(x)=\log _{5}(x+6) $$

The functions $$ f(x)=6.43(1.027)^{x} \text { and } g(x)=\frac{40.9}{1+6.6 e^{-0.04 x}} $$ model the percentage of college graduates among people ages 25 and older, \(f(x)\) or \(g(x), x\) years after \(1950 .\) Use these functions. Which function is a better model for the percentage who were college graduates in 1990?

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In \(2006,\) Canada's population exceeded Uganda's by 4.9 million.

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{\pi} 63 $$

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. $$ \log (x+4)=\log x+\log 4 $$

In Exercises 75–80, find the domain of each logarithmic function. $$ f(x)=\log (2-x) $$

What is an exponential function?

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. By \(2009,\) the models indicate that Canada's population will exceed Uganda's by approximately 2.8 million.

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{\pi} 400 $$

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. $$ \log (5 x+1)=\log (2 x+3)+\log 2 $$

In Exercises 75–80, find the domain of each logarithmic function. $$ f(x)=\log (7-x) $$

What is the natural exponential function?

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The models indicate that in 2013 , Uganda's population will exceed Canada's.

Use a graphing utility and the change-of-base property to graph each function. $$ y=\log _{3} x $$

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. $$ \log (3 x-3)=\log (x+1)+\log 4 $$

In Exercises 75–80, find the domain of each logarithmic function. $$ f(x)=\ln (x-2)^{2} $$

Use a calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for \(x=10,100,1000\) \(10,000,100,000,\) and \(1,000,000 .\) Describe what happens to the expression as \(x\) increases.

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Uganda's growth rate is approximately 3.8 times that of Canada's.

Use a graphing utility and the change-of-base property to graph each function. $$ y=\log _{15} x $$

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

In Exercises 75–80, find the domain of each logarithmic function. $$ f(x)=\ln (x-7)^{2} $$

Use a graphing utility and the change-of-base property to graph each function. $$ y=\log _{2}(x+2) $$

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. $$ 2 \log x=\log 25 $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \log 100 $$.

You have \(\$ 10,000\) to invest. One bank pays \(5 \%\) interest compounded quarterly and a second bank pays \(4.5 \%\) interest compounded monthly. a. Use the formula for compound interest to write a function for the balance in each bank at any time \(t .\) b. Use a graphing utility to graph both functions in an appropriate viewing rectangle. Based on the graphs, which bank offers the better return on your money?

Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by exponential or logarithmic functions. Group members should select the two sets of data that are most interesting and relevant. For each set selected, find a model that best fits the data. Each group member should make one prediction based on the model and then discuss a consequence of this prediction. What factors might change the accuracy of the prediction?

Use a graphing utility and the change-of-base property to graph each function. $$ y=\log _{3}(x-2) $$

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. $$ 3 \log x=\log 125 $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \log 1000 $$

a. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) in the same viewing rectangle. b. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) in the same viewing rectangle. c. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing rectangle. d. Describe what you observe in parts (a)-(c). Try generalizing this observation.

We have shown that the graph of the logarithmic function \(y=\log _{3}(x-\) can be ploted in the viewing rectangle when the logarithmic functio \(\log _{3}(x-2)\) can be written in terms of natural logarithmic function using change of base property, as follows $$y=\log _{3}(x-2)=\frac{\ln (x-2)}{\ln (3)}$$.

After a \(60 \%\) price reduction, you purchase a computer for \(\$ 440 .\) What was the computer's price before the reduction?

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\)Write each expression in terms of \(A\) and \(C\). $$ \log _{b} \frac{3}{2} $$