Chapter 8

Trisha said that the equation \(5^{x}+6=127\) could be solved by writing the logarithmic equation \(x \log 5+\log 6=\log 127 .\) Do you agree with Trisha? Explain why or why not.

Randall said that the equation \(\log x+\log 12=\log 9\) can be solved by writing the equation \(x+12=9 .\) Do you agree with Randall? Explain why or why not.

Explain why \(\log 80=1+\log 8\)

If \(\log _{b} c=a,\) explain why \(\log _{b} \frac{1}{c}=-a\)

Show that for all \(a>0\) and \(a \neq 1, \log _{a} a^{n}=n\)

Peg said that \((1,0)\) is always a point on the graph of \(y=\log _{b} x .\) Do you agree with Peg? Explain why or why not.

Melita said that the equation \(4(3)^{x}=72\) could be solved by writing the logarithmic equation \(x \log 12=\log 72 .\) Do you agree with Melita? Explain why or why not.

For what value of \(a\) does \(\log a=\ln a ?\) Justify your answer.

Pritha said that before an equation such as \(\log x=1+\log 5\) can be solved, 1 could be written as \(\log 10 .\) Do you agree with Pritha? Explain why or why not.

Explain why log \(x\) is negative if \(0 < x < 1\)

If \(\log _{b} c=a,\) explain why \(\log _{b} c^{2}=2 a\)

Terence said that \(\left(\log _{a} b\right) \cdot\left(\log _{a} c\right)=\log _{a} b c .\) Do you agree with Terence? Explain why or why not.

Sue said that if \(x=b^{2 y}\) for \(b>1,\) then \(y=\frac{1}{2} \log _{b} x .\) Do you agree with Sue? Explain why or why not.

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ 3.75 $$

In \(3-14,\) solve each equation for the variable. Express each answer to the nearest hundredth. $$ 3^{x}=12 $$

Solve each equation for the variable and check. \(\log x+\log 8=\log 200\)

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 3.75 $$

In \(3-14,\) write each exponential equation in logarithmic form. $$ 2^{4}=16 $$

\(\ln 3-10 :\) a. For each \(f(x),\) write an equation for \(f^{-1}(x),\) the inverse function. b. Sketch the graph of \(f(x)\) and of \(f^{-1}(x) .\) $$ f(x)=3^{x} $$

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ 8.56 $$

In \(3-14,\) solve each equation for the variable. Express each answer to the nearest hundredth. $$ 2^{b}=18 $$

Solve each equation for the variable and check. \(\log x+\log 15=\log 90\)

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 8.56 $$

In \(3-14,\) write each exponential equation in logarithmic form. $$ 5^{3}=125 $$

\(\ln 3-10 :\) a. For each \(f(x),\) write an equation for \(f^{-1}(x),\) the inverse function. b. Sketch the graph of \(f(x)\) and of \(f^{-1}(x) .\) $$ \mathrm{f}(x)=5^{x} $$

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ 47.88 $$

In \(3-14,\) solve each equation for the variable. Express each answer to the nearest hundredth. $$ 5^{y}=100 $$

Solve each equation for the variable and check. \(\ln x+\ln 18=\ln 27\)

In \(3-14,\) write each exponential equation in logarithmic form. $$ 64=8^{2} $$

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 47.88 $$

\(\ln 3-10 :\) a. For each \(f(x),\) write an equation for \(f^{-1}(x),\) the inverse function. b. Sketch the graph of \(f(x)\) and of \(f^{-1}(x) .\) $$ \mathrm{f}(x)=\left(\frac{3}{2}\right)^{x} $$

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ 56.2 $$

In \(3-14,\) solve each equation for the variable. Express each answer to the nearest hundredth. $$ 10^{x}=50 $$

Solve each equation for the variable and check. \(\log x-\log 5=\log 6\)

In \(3-14,\) write each exponential equation in logarithmic form. $$ 12^{0}=1 $$

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 56.2 $$

\(\ln 3-10 :\) a. For each \(f(x),\) write an equation for \(f^{-1}(x),\) the inverse function. b. Sketch the graph of \(f(x)\) and of \(f^{-1}(x) .\) $$ f(x)=\left(\frac{5}{2}\right)^{x} $$

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ 562 $$

In \(3-14,\) solve each equation for the variable. Express each answer to the nearest hundredth. $$ 12^{a}=254 $$

Solve each equation for the variable and check. \(\log x-\log 3=\log 42\)

In \(3-14,\) write each exponential equation in logarithmic form. $$ 216=6^{3} $$

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 562 $$

\(\ln 3-10 :\) a. For each \(f(x),\) write an equation for \(f^{-1}(x),\) the inverse function. b. Sketch the graph of \(f(x)\) and of \(f^{-1}(x) .\) $$ f(x)=\left(\frac{1}{2}\right)^{x} $$

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ 5,620 $$

In \(3-14,\) solve each equation for the variable. Express each answer to the nearest hundredth. $$ 6\left(3^{x}\right)=532 $$

Solve each equation for the variable and check. \(\ln x-\ln 24=\ln 8\)

In \(3-14,\) write each exponential equation in logarithmic form. $$ 10^{-1}=0.1 $$

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 5,620 $$

\(\ln 3-10 :\) a. For each \(f(x),\) write an equation for \(f^{-1}(x),\) the inverse function. b. Sketch the graph of \(f(x)\) and of \(f^{-1}(x) .\) $$ f(x)=(\sqrt{2})^{x} $$

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ 0.342 $$