Chapter 12 - Exponential Functions and Logarithmic Functions - 12.3 Logarithmic Functions - 12.3 Exercise Set - Page 803: 101

The graph is shown below.

$y={{\left( \frac{3}{2} \right)}^{x}}$ and $y={{\log }_{\frac{3}{2}}}x$ Consider the first equation, $y={{\left( \frac{3}{2} \right)}^{x}}$ In order to draw the graph, substitute $x=-2,-1,0,1,2$ in the function $y={{\left( \frac{3}{2} \right)}^{x}}$ one by one. Substitute in a selected x value: $\begin{align} & y={{\left( \frac{3}{2} \right)}^{-2}} \\ & ={{\left( \frac{2}{3} \right)}^{2}} \\ & =\frac{{{2}^{2}}}{{{3}^{2}}} \\ & =\frac{4}{9} \end{align}$ Substitute in a selected x value: $\begin{align} & y={{\left( \frac{3}{2} \right)}^{-1}} \\ & ={{\left( \frac{2}{3} \right)}^{1}} \\ & =\frac{2}{3} \end{align}$ Substitute in a selected x value: $\begin{align} & y={{\left( \frac{3}{2} \right)}^{0}} \\ & =1 \end{align}$ Substitute in a selected x value: $\begin{align} & y={{\left( \frac{3}{2} \right)}^{1}} \\ & =\frac{3}{2} \end{align}$ Substitute in a selected x value: $\begin{align} & y={{\left( \frac{3}{2} \right)}^{2}} \\ & =\frac{{{3}^{2}}}{{{2}^{2}}} \\ & =\frac{9}{4} \end{align}$ $\begin{matrix} x & y={{\left( \frac{3}{2} \right)}^{x}} \\ 0 & 1 \\ 1 & \frac{3}{2} \\ 2 & \frac{9}{4} \\ -1 & \frac{2}{3} \\ -2 & \frac{4}{9} \\ Table 1 \end{matrix}$ Now, consider the second equation, $y={{\log }_{\frac{3}{2}}}x$ The function $y={{\log }_{\frac{3}{2}}}x$ can be written as ${{\left( \frac{3}{2} \right)}^{y}}=x$. In order to draw the graph, substitute $y=0,1,2,-1,-2$ in the function $x={{\left( \frac{3}{2} \right)}^{y}}$. Substitute in a selected x value: $\begin{align} & x={{\left( \frac{3}{2} \right)}^{-2}} \\ & ={{\left( \frac{2}{3} \right)}^{2}} \\ & =\frac{{{2}^{2}}}{{{3}^{2}}} \\ & =\frac{4}{9} \end{align}$ Substitute in a selected x value: $\begin{align} & x={{\left( \frac{3}{2} \right)}^{-1}} \\ & ={{\left( \frac{2}{3} \right)}^{1}} \\ & =\frac{2}{3} \end{align}$ Substitute in a selected x value: $\begin{align} & x={{\left( \frac{3}{2} \right)}^{0}} \\ & =1 \end{align}$ Substitute in a selected x value: $\begin{align} & x={{\left( \frac{3}{2} \right)}^{1}} \\ & =\frac{3}{2} \end{align}$ Substitute in a selected x value: $\begin{align} & x={{\left( \frac{3}{2} \right)}^{2}} \\ & =\frac{{{3}^{2}}}{{{2}^{2}}} \\ & =\frac{9}{4} \end{align}$ $\begin{matrix} x & x={{\left( \frac{3}{2} \right)}^{y}} \\ 0 & 1 \\ 1 & \frac{3}{2} \\ 2 & \frac{9}{4} \\ -1 & \frac{2}{3} \\ -2 & \frac{4}{9} \\ Table 2 \end{matrix}$ Plot these points from table 1 and 2 and connect them with a smooth curve as shown below.