Chapter 6.4, Problem 51E

To determine

To explain why the property of Equality for Exponential Equations does not work with $b=1$

Also give example to show the same.

Explanation of Solution

Given information:

In an exponential function $b=1$

Concept used:

Standard equation of exponential function is

  $y=a{b}^x\text{ where }a\ne 0$

If $b=1$ , then standard equation reduces to

  $y=a{\left(1\right)}^x$

For any real value of $x$

  ${\left(1\right)}^x=1$

The function becomes $y=a$

This is no longer an exponential function,

So,

  $b=1$ is not possible in case of exponential function $y=a{b}^x\text{ where }a\ne 0$

For example,

For $a=2\text{ and }b=1$

Exponential function is

  $\begin{array}{l}y=2{\left(1\right)}^x\\ \Rightarrow y=2\times 1=2\end{array}$

That is constant not an exponential function.

Hence,

  $b\ne 1$ for exponential function