Chapter 6.6, Problem 4E

To determine

Fill in the blank and make the statement correct.

Answer to Problem 4E

If $b$ is a positive real number other than $1$ , then $b^x=b^y$ if and only if $x=y$ .

Explanation of Solution

Given Information:

The given statement is:

If $b$ is a positive real number other than $1$ , then $b^x=b^y$ if and only if ___________.

Formula Used Power rule of logarithmic functions:

  ${\log }_a{x}^p=p{\log }_{a}x$

Where $a$ is base, $x$ is the variables and $p$ is power of variables.

Calculation:

Consider the equation is $b^x=b^y$ .

Take common logarithm with base $10$ of the above equation.

  ${\log }_{10}{b}^x={\log }_{10}{b}^y$

Now, use power rule of logarithm ${\log }_a{x}^p=p{\log }_{a}x$ in the above equation.

  $x{\log }_{10}b=y{\log }_{10}b$

Divide each side by ${\log }_{10}b$ in the above equations.

  $\begin{array}{c}\frac{x{\log }_{10}b}{{\log }_{10}b}=\frac{y{\log }_{10}b}{{\log }_{10}b}\\ x=y\end{array}$

Hence, if $b$ is a positive real number other than $1$ , then $b^x=b^y$ if and only if $x=y$ .