Question

67. CRITICAL THINKING Decide whether the following statement is sometimes, always, or never true. Explain your reasoning.

For a positive base b other than 1, $b^x>b^y$. if and only if $x>y$.

Short answer

The given statement is sometimes true.

Step-by-step solution

Step 1.   Given

For a positive base b other than 1, $b^x>b^y$ if and only if $x>y$.

Step 2.   To determine

Decide whether the following statement is sometimes, always, or never true. Explain your reasoning.

Step 3.  Calculation

For $b=2,x=4,y=3$

$\begin{array}{l}{b}^x>b^y\\ 2^4>2^3\\ 16>8\end{array}$

This is true.

For $b=\frac{1}{2},x=4,y=3$

$\begin{array}{l}{b}^x>b^y\\ {\left(\frac{1}{2}\right)}^4>{\left(\frac{1}{2}\right)}^3\\ \frac{1}{16}>\frac{1}{8}\\ 8>16\end{array}$

This is not true.

So, the given statement is sometimes true.