Chapter 4

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ 3 \log _{2}(x-1)=5-\log _{2} 4 $$

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{16} 57.2 $$

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. When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative.

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log _{2}(x-6)+\log _{2}(x-4)-\log _{2} x=2 $$

Find the domain of each logarithmic function. $$ f(x)=\log _{5}(x+6) $$

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.

Solve each logarithmic equation in Exercises \(49-92 .\) 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 \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{0.3} 19 $$

Find the domain of each logarithmic function. $$ f(x)=\log (2-x) $$

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log (x+4)=\log x+\log 4 $$

What is an exponential function?

Find the domain of each logarithmic function. $$ f(x)=\log (7-x) $$

The exponential growth models describe the population of the indicated country, \(A,\) in millions, \(t\) years after 2006 $$\begin{aligned}&\text { Carada } \quad A=33.1 e^{0.000 t}\\\&\text { Ugaria } \quad A=28.2 e^{0.034 t}\end{aligned}$$ In Exercises \(77-80\), 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.

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log (5 x+1)=\log (2 x+3)+\log 2 $$

What is the natural exponential function?

Find the domain of each logarithmic function. $$ f(x)=\ln (x-2)^{2} $$

The exponential growth models describe the population of the indicated country, \(A,\) in millions, \(t\) years after 2006 $$\begin{aligned}&\text { Carada } \quad A=33.1 e^{0.000 t}\\\&\text { Ugaria } \quad A=28.2 e^{0.034 t}\end{aligned}$$ In Exercises \(77-80\), 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

Solve each logarithmic equation in Exercises \(49-92 .\) 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 \(79-82,\) use a graphing utility and the change-of-base property to graph each function. $$ y=\log _{3} x $$

Use a calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for \(x-10,100,1000\) \(10,000,100,000,\) and \(1,000,009 .\) Describe what happens to the expression as \(x\) increases

Find the domain of each logarithmic function. $$ f(x)=\ln (x-7)^{2} $$

The exponential growth models describe the population of the indicated country, \(A,\) in millions, \(t\) years after 2006 $$\begin{aligned}&\text { Carada } \quad A=33.1 e^{0.000 t}\\\&\text { Ugaria } \quad A=28.2 e^{0.034 t}\end{aligned}$$ In Exercises \(77-80\), 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

Solve each logarithmic equation in Exercises \(49-92 .\) 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 \(79-82,\) use a graphing utility and the change-of-base property to graph each function. $$ y=\log _{15} x $$

Evaluate or simplify each expression without using a calculator. $$ \log 100 $$

Solve each logarithmic equation in Exercises \(49-92 .\) 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 \(79-82,\) use a graphing utility and the change-of-base property to graph each function. $$ y=\log _{2}(x+2) $$

You have \(\$ 10,000\) to inves \(\$$ One bank pays \)5 \%\( interest compounded quarterly and sisccond bank pays \)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?

Evaluate or simplify each expression without using a calculator. $$ \log 1000 $$

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 cach 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?

Solve each logarithmic equation in Exercises \(49-92 .\) 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 \(79-82,\) use a graphing utility and the change-of-base property to graph each function. $$ y=\log _{3}(x-2) $$

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.

Evaluate or simplify each expression without using a calculator. $$ \log 10^{7} $$

Exercises \(83-85\) will help you prepare for the material covered in the first section of the next chapter. a. Does \((4,-1)\) satisfy \(x+2 y-2 ?\) b. Does \((4,-1)\) satisfy \(x-2 y-6 ?\)

In Exercises \(83-88,\) let \(\log _{b} 2=A\) and \(\log _{b} 3=\) C. Write each expression in terms of \(A\) and \(C\). $$ \log _{b} \frac{3}{2} $$

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log (x+4)-\log 2=\log (5 x+1) $$

Evaluate or simplify each expression without using a calculator. $$ \log 10^{8} $$

Exercises \(83-85\) will help you prepare for the material covered in the first section of the next chapter. Graph \(x+2 y-2\) and \(x-2 y-6\) in the same rectangular coordinate system. At what point do the graphs intersect?

In Exercises \(83-88,\) let \(\log _{b} 2=A\) and \(\log _{b} 3=\) C. Write each expression in terms of \(A\) and \(C\). $$ \log _{b} 6 $$

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log (x+7)-\log 3=\log (7 x+1) $$

Make Sense? In Exercises \(83-86,\) determine whether catch statement makes sense or does not make sense, and explain your reasoning. I'm using a photocopier to reduce an image over and over by \(50 \%,\) so the exponential function \(f(x)-\left(\frac{1}{2}\right)^{x}\) models the new image size, where \(x\) is the number of reductions.

Evaluate or simplify each expression without using a calculator. $$ 10^{\log 33} $$

Exercises \(83-85\) will help you prepare for the material covered in the first section of the next chapter. $$\text { Solve: } 5(2 x-3)-4 x-9$$

In Exercises \(83-88,\) let \(\log _{b} 2=A\) and \(\log _{b} 3=\) C. Write each expression in terms of \(A\) and \(C\). $$ \log _{b} 8 $$

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ 2 \log x-\log 7=\log 112 $$

I'm using a photocopier to reduce an image over and over by \(50 \%,\) so the exponential function \(f(x)-\left(\frac{1}{2}\right)^{x}\) models the new image size, where \(x\) is the number of reductions. I'm looking at data that show the number of new college programs in green studies, and a linear function appears to be a better choice than an exponential function for modeling the number of new college programs from 2005 through 2009 .

Evaluate or simplify each expression without using a calculator. $$ 10^{\log 53} $$

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log (x-2)+\log 5=\log 100 $$