Problem 95 - Introductory and Intermediate Algebra for College Students 4th

Question

What question can be asked to help evaluate \(\log _{3} 81 ?\)

Step-by-Step Solution

Answer

The value of \(\log _{3} 81\) is 4.

Step 1: Understanding Logarithms

Firstly, a logarithm is simply an exponent applied to a specific base to obtain a number. In this example, we're asking the question '3 raised to what power gives us 81?'

Step 2: Evaluating the Logarithm

We can simplify this expression by expressing 81 as a power of 3. We know that \(3^4 = 81\). Looking at this, we can see that 4 is the power that we raise 3 to, to obtain 81.

Step 3: Associate the Logarithm

Then let's apply this to the logarithmic expression. We may rewrite our original log expression; \(\log _{3} 81\) as \(\log _{3} (3^4)\). Now, because the logarithm and the exponent are inverse operations, they cancel each other out. Hence the solution to \(\log _{3} (3^4)\) is 4