Chapter 4.3, Problem 46E

To determine

To find: The function of the $y={\log }_{a}x$ .

Answer to Problem 46E

The logarithm function of the given graph is ${\log }_{2}x=y$ .

Explanation of Solution

Given:

The given graph is shown in Figure 1

Figure 1

Calculation:

Consider the points on the graph is,

  $\left(x,y\right)=\left(\frac{1}{2},-1\right)$

Consider the logarithm function is,

  ${\log }_{a}x=y$

Then, the exponential form of the above equation is,

  $a^y=x$

From the given points $\left(x,y\right)=\left(\frac{1}{2},-1\right)$ the logarithm function of the given graph is,

  ${\log }_a\frac{1}{2}=-1$

Then the exponential form of the above expression is,

  $\begin{array}{l}{a}^1=\frac{1}{2}\\ \frac{1}{a}=\frac{1}{2}\\ a=2\end{array}$

Then, the required logarithm function is,

  ${\log }_{2}x=y$