Chapter 3.2, Problem 130E

To determine

To describe : the error.

Answer to Problem 130E

  $y=2^x$

So, y is an exponential function of x , and not a logarithmic function.

Explanation of Solution

Given information :

The table:

  

From the table, it can be concluded that y is a logarithmic function of x .

As per the problem,

Consider the given table.

If y is a logarithmic function of x , then the table of values should satisfy the equation

  $y={\log }_{a}x$

Substituting any value of ( x, y ) from the table in the equation to find the value of the base a .

  $\begin{array}{l}32={\log }_{a}5\text{ [Take }x=5,y=32]\\ a^{32}=5\end{array}$

No integer raised to the power $32$ is equal to $5$

Hence, y is not a logarithmic function of x.

However, if y is an exponential function of x , then

  $y=a^x$

For $x=5$ and $y=32$ ,

  $\begin{array}{l}32=a^5\\ 2^5=a^5\\ 2=a\end{array}$

Taking the exponentialas $2$ ,

  

Therefore, $y=2^x$ satisfies all the values of the table.

Hence, y is an exponential function of x , and not a logarithmic function.