Problem 87 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\log _{3} 7=\frac{1}{\log _{7} 3}$$

Step-by-Step Solution

Answer

The given equation \(\log _{3} 7=\frac{1}{\log _{7} 3}\) is true, as both sides of the equation evaluate to the same value.

Step 1: Simplify Both Sides of the Equation

Start from the equation, \(\log _{3} 7=\frac{1}{\log _{7} 3}\), a cornerstone property used here is that \(\log_b a = 1/ \log_a b\). So \(\log _{3} 7\) produces the same result as \(\frac{1}{\log _{7} 3}\). Both sides of the equation evaluate to the same value, it's clear the equation is true.

Step 2: Test by Substituting Values

To be sure of the validity of the equation, substitute values. Using a calculator, \(\log _{3} 7\) equals approximately 1.77124, and \(\frac{1}{\log _{7} 3}\) also gives approximately 1.77124. Thus confirming that the initial equation is true.

Step 3: Confirm the Truth of the Equation

Since both sides of the equation evaluate to the same value, and substituting values produces the same results on both sides, it can be concluded that the equation \(\log _{3} 7=\frac{1}{\log _{7} 3}\) is true. There's no need to make any changes to make a true statement because it's already true.