Chapter 10

Use this approach to find the inverse of each of the following functions. See below. (a) \(f(x)=3 x-9\) (b) \(f(x)=-2 x+6\) (c) \(f(x)=-x+1\) (d) \(f(x)=2 x\) (e) \(f(x)=-5 x\) (f) \(f(x)=x^{2}+6\) for \(x \geq 0\) If \(f(x)=2 x+3\) and \(g(x)=3 x-5\), find (a) \((f \circ g)^{-1}(x) \frac{x+7}{6}\) (b) \(\left(f^{-1} \circ g^{-1}\right)(x)\) (c) \(\left(g^{-1} \circ f^{-1}\right)(x)\) \(\frac{x-4}{6}\)

Use both a graphical and an algebraic approach to solve the equation \(\frac{2^{x}-2^{-x}}{3}=4\).

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} y^{3} z^{4} $$

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} x^{2} y^{3} $$

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b}\left(\frac{x^{1 / 2} y^{1 / 3}}{z^{4}}\right) $$

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} x^{2 / 3} y^{3 / 4} $$

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} \sqrt[3]{x^{2} z} $$

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} \sqrt{x y} $$

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b}\left(x \sqrt{\frac{x}{y}}\right) $$

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} \sqrt{\frac{x}{y}} $$

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ 2 \log _{b} x-4 \log _{b} y $$

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ \log _{b} x+\log _{b} y-\log _{b} z $$

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ \log _{b} x-\left(\log _{b} y-\log _{b} z\right) $$

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ \left(\log _{b} x-\log _{b} y\right)-\log _{b} z $$

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ 2 \log _{b} x+4 \log _{b} y-3 \log _{b} z $$

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ \log _{b} x+\frac{1}{2} \log _{b} y $$

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ \frac{1}{2} \log _{b} x-\log _{b} x+4 \log _{b} y $$

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ 2 \log _{b} x+\frac{1}{2} \log _{b}(x-1)-4 \log _{b}(2 x+5) $$

Solve each equation. $$ \log _{3} x+\log _{3} 4=2 $$

Solve each equation. $$ \log _{7} 5+\log _{7} x=1 $$

Solve each equation. $$ \log _{10} x+\log _{10}(x-21)=2 $$

Solve each equation. $$ \log _{10} x+\log _{10}(x-3)=1 $$

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

Solve each equation. $$ \log _{3} x+\log _{3}(x-2)=1 $$

Solve each equation. $$ \log _{3}(x+3)+\log _{3}(x+5)=1 $$

Solve each equation. $$ \log _{2}(x+2)=1-\log _{2}(x+3) $$

Solve each equation. $$ \log _{2} 3+\log _{2}(x+4)=3 $$

Solve each equation. $$ \log _{4} 7+\log _{4}(x+3)=2 $$

Solve each equation. $$ \log _{10}(2 x-1)-\log _{10}(x-2)=1 $$

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

Solve each equation. $$ \log _{6} x+\log _{6}(x+5)=2 $$

Solve each equation. $$ \log _{2}(x-1)-\log _{2}(x+3)=2 $$

Solve each equation. $$ \log _{5} x=\log _{5}(x+2)+1 $$

Solve each equation. $$ \log _{8}(x+7)+\log _{8} x=1 $$

Solve each equation. $$ \log _{6}(x+1)+\log _{6}(x-4)=2 $$

Explain, without using Property \(10.4\), why \(4^{\log _{4} 9}\) equals \(9 .\)

How would you explain the concept of a logarithm to someone who had just completed an elementary algebra course?

Solve each equation. $$ \log _{10}(9 x-2)=1+\log _{10}(x-4) $$