Problem 34 - College Algebra Essentials

Question

Evaluate each expression without using a calculator. $$\log _{81} 9$$

Step-by-Step Solution

Answer

The result of \(\log _{81} 9\) is 0.5

Step 1: Recognize the Base and the Argument

It's key to note that both 9 and 81 are powers of 3. \(81 = 3^{4}\) and \(9 = 3^{2}\), this enables us to express the expression in a more familiar base for simplification.

Step 2: Employ the Change of Base Formula

The Change of Base Formula states that for \(log _{b} a\), it can be expressed as \(\frac{log _{c} a}{log _{c} b}\) where c is often 10 in a calculator or in cases like this, a number where the result can be known readily, in this case, 3. Hence \(log _{81} 9\) can be expressed as \(\frac{log _{3} 9}{log _{3} 81}\).

Step 3: Deduce the Value

Given that \(9 = 3^{2}\) and \(81 = 3^{4}\), it applies that \(log _{3} 9 = 2\) and \(log _{3} 81 = 4\). Thus, \(\frac{log _{3} 9}{log _{3} 81}\) equals \(\frac{2}{4} = 0.5\)