Chapter 12

If \(\log _{b} 3=0.5\) and \(\log _{b} 5=0.7,\) evaluate each expression. $$ \log _{b} \sqrt{5} $$

Solve \(5^{x}=9\) by taking the common logarithm of both sides of the equation. Next, solve this equation by taking the natural logarithm of both sides. Compare your solutions. Are they the same? Why or why not?

Solve. $$ \log _{2} 16=x $$

Approximate each logarithm to four decimal places. $$ \log _{6} \frac{2}{3} $$

Describe the appearance of the graphs of a function and its inverse.

If \(\log _{b} 3=0.5\) and \(\log _{b} 5=0.7,\) evaluate each expression. $$ \log _{b} \sqrt[4]{3} $$

Use a graphing calculator to solve. Round your answers to two decimal places. $$ e^{0.3 x}=8 $$

Solve. $$ \log _{3 / 4} x=3 $$

Approximate each logarithm to four decimal places. $$ \log _{1 / 2} 5 $$

Find the inverse of each one-to-one function. Then graph the function and its inverse in a square window. $$ f(x)=3 x+1 $$

If \(\log _{b} 2=0.43\) and \(\log _{b} 3=0.68\), evaluate each expression. $$ \log _{b} 8 $$

Use a graphing calculator to solve. Round your answers to two decimal places. $$ 10^{0.5 x}=7 $$

Solve. $$ \log _{2 / 3} x=2 $$

Approximate each logarithm to four decimal places. $$ \log _{1 / 3} 2 $$

Find the inverse of each one-to-one function. Then graph the function and its inverse in a square window. $$ f(x)=-2 x-6 $$

If \(\log _{b} 2=0.43\) and \(\log _{b} 3=0.68\), evaluate each expression. $$ \log _{b} 81 $$

Solve. $$ \log _{x} 100=2 $$

Solve for \(x\). $$ 6 x-3(2-5 x)=6 $$

Find the inverse of each one-to-one function. Then graph the function and its inverse in a square window. $$ f(x)=\sqrt[3]{x+3} $$

Explain why the graph of an exponential function \(y=b^{x}\) contains the point \((1, b)\).

If \(\log _{b} 2=0.43\) and \(\log _{b} 3=0.68\), evaluate each expression. $$ \log _{b} \frac{3}{9} $$

Solve. $$ \log _{x} 27=3 $$

Solve for \(x\). $$ 2 x+3=5-2(3 x-1) $$

Find the inverse of each one-to-one function. Then graph the function and its inverse in a square window. $$ f(x)=x^{3}-3 $$

Explain why an exponential function \(y=b^{x}\) has a \(y\) -intercept of (0,1)

If \(\log _{b} 2=0.43\) and \(\log _{b} 3=0.68\), evaluate each expression. $$ \log _{b} \frac{4}{32} $$

Solve. $$ \log _{2} 2^{4}=x $$

If \(\log _{b} 2=0.43\) and \(\log _{b} 3=0.68\), evaluate each expression. $$ \log _{b} \sqrt{\frac{2}{3}} $$

Solve. $$ \log _{6} 6^{-2}=x $$

If \(\log _{b} 2=0.43\) and \(\log _{b} 3=0.68\), evaluate each expression. $$ \log _{b} \sqrt{\frac{3}{2}} $$

Solve. $$ 3^{\log _{3} 5}=x $$

Solve for \(x\). $$ x^{2}+7 x=-6 $$

Solve. $$ 5^{\log _{8} 7}=x $$

Solve for \(x\). $$ x^{2}+4 x=12 $$

Solve. $$ \log _{x} \frac{1}{7}=\frac{1}{2} $$

Use a calculator to try to approximate \(\log 0\). Describe what happens and explain why.

Evaluate each expression. $$ \log _{10} 100 $$

Solve. $$ \log _{x} 2=-\frac{1}{3} $$

Use a calculator to try to approximate \(\ln 0 .\) Describe what happens and explain why.

Evaluate each expression. $$ \log _{10} \frac{1}{10} $$

Simplify. $$ \log _{5} 5^{3} $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{x} $$

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

Simplify. $$ \log _{6} 6^{2} $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{2 x} $$

Evaluate each expression. $$ \log _{7} \sqrt{7} $$

Simplify. $$ 2^{\log _{2} 3} $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=\ln x $$

Solve. Which of the following is the correct way to rewrite \(\log _{3} \frac{14}{11} ?\) a. \(\frac{\log _{3} 14}{\log _{3} 11}\) b. \(\log _{3} 14-\log _{3} 11\) c. \(\log _{3}(14-11)\) d. \(\log _{3} 154\)

Simplify. $$ 7^{\log _{7} 4} $$