Chapter 4

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. $$ \log (x+4)-\log 2=\log (5 x+1) $$

Begin by graphing \(y=|x| .\) Then use this graph to obtain the graph of \(y=|x-2|+1 . \)

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. 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. $$ \log (x+7)-\log 3=\log (7 x+1) $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \log 10^{8} $$

determine whether each 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.

Write an equation in point-slope form and slope-intercept form of the line passing through \((1,-4)\) and parallel to the line whose equation is \(3 x-y+5=0 .\)

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. 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. $$ 2 \log x-\log 7=\log 112 $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ 10^{\log 33} $$

determine whether each statement makes sense or does not make sense, and explain your reasoning. Taxing thoughts: I'm looking at data that show the number of pages in the publication that explains the U.S. tax code for selected years from 1945 thorugh 2013 . A linear function appears to be a better choice than an exponential function for modeling the number of pages in the tax code during this period.

Exercises \(86-88\) will help you prepare for the material covered in the first section of the next chapter. $$ \text { Solve: } \quad \frac{5 \pi}{4}=2 \pi x $$

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\)Write each expression in terms of \(A\) and \(C\). $$ \log _{b} 81 $$

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. $$ \log (x-2)+\log 5=\log 100 $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ 10^{\log 53} $$

determine whether each statement makes sense or does not make sense, and explain your reasoning. I use the natural base \(e\) when determining how much money I'd have in a bank account that earns compound interest subject to continuous compounding.

Exercises \(86-88\) will help you prepare for the material covered in the first section of the next chapter. $$ \text { Simplify: } \frac{17 \pi}{6}-2 \pi $$

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\)Write each expression in terms of \(A\) and \(C\). $$ \log _{b} \sqrt{\frac{2}{27}} $$

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. $$ \log x+\log (x+3)=\log 10 $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln 1 $$

Exercises \(86-88\) will help you prepare for the material covered in the first section of the next chapter. $$ \text { Simplify: }-\frac{\pi}{12}+2 \pi $$

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\)Write each expression in terms of \(A\) and \(C\). $$ \log _{b} \sqrt{\frac{3}{16}} $$

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. $$ \log (x+3)+\log (x-2)=\log 14 $$

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

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. $$ \ln (x-4)+\ln (x+1)=\ln (x-8) $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln e^{20} $$

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. $$ \log _{2}(x-1)-\log _{2}(x+3)=\log _{2}\left(\frac{1}{x}\right) $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln e^{7} $$

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The functions \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\) have the same graph.

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. $$ \log _{4}\left(2 x^{3}\right)=3 \log _{4}(2 x) $$

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. $$ \ln (x-2)-\ln (x+3)=\ln (x-1)-\ln (x+7) $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln \frac{1}{e^{6}} $$

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln \frac{1}{e^{7}} $$

Graph \(f(x)=2^{x}\) and its inverse function in the same rectangular coordinate system.

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

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ e^{\ln 125} $$

The hyperbolic cosine and hyperbolic sine functions are defined by $$ \cosh x=\frac{e^{x}+e^{-x}}{2} \quad \text { and } \quad \sinh x=\frac{e^{x}-e^{-x}}{2} $$ a. Show that \(\cosh x\) is an even function. b. Show that \(\sinh x\) is an odd function. c. Prove that \((\cosh x)^{2}-(\sinh x)^{2}=1\)

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

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

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ e^{\ln 300} $$

Solve for \(y: 7 x+3 y=18 .\) (Section \(1.3,\) Example 7 )

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

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

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln e^{9 x} $$

Find all zeros of \(f(x)=x^{3}+5 x^{2}-8 x+2 .\) (Section 3.4 Example 4 )

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

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln e^{13 x} $$

Solve and graph the solution set on a number line: \(2 x^{2}+5 x<12\). (Section \(3.6,\) Example 1 )

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

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ e^{\ln 5 x^{2}} $$