Chapter 8.4, Problem 40HP

To determine

The explanation of the reasoning.

Answer to Problem 40HP

When $0<b<1,{\log }_{b}x>{\log }_{b}y$ if and only if $x<y$ . The inequality symbol is switched because a fraction that is less than $1$ becomes smaller when it is taken to a greater power.

Explanation of Solution

Given information:

The property of Inequality for logarithmic functions states that when $b>1,{\log }_{b}x>{\log }_{b}y$ if and only if $x>y$ .What is the case for when $0<b<1?$ Explain your reasoning.

Calculation:

When $0<b<1,{\log }_{b}x>{\log }_{b}y$ if and only if $x<y$ . The inequality symbol is switched because a fraction that is less than $1$ becomes smaller when it is taken to a greater power.