Problem 121 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The question \(\log_{3}{81}\) simply asks: 'To what power must we raise 3 to get 81?'. The answer is 4, because \(3^4 = 81\).

Step 1: Understanding Logarithms

A logarithm has a base (here it's 3), and a number (here it's 81). The logarithm \(\log_{b}{a}\) asks what power you must raise the base \(b\) to in order to get \(a\). Thus, \(\log_{3}{81}\) means: 'what power should 3 be raised to in order to get 81?'.

Step 2: Apply Logarithm Properties

If the expression is written in exponential form such as \(3^n = 81\), we can now solve for \(n\). The question is the equivalent of finding \(n\) in that exponential equation.

Step 3: Solve for n

With the base being 3, we want to express 81 in terms of 3. \(81 = 3^4\). Therefore if we know \(3^n = 3^4\) we can compare the exponents and determine \(n = 4\).