Problem 26 - College Algebra Essentials

Question

Evaluate each expression without using a calculator. $$\log _{6} \frac{1}{6}$$

Step-by-Step Solution

Answer

\(\log _{6} \frac{1}{6} = -1\)

Step 1: Identify the base and antilogarithm

In the given expression \(\log _{6} \frac{1}{6}\), the base of the logarithm is 6 and its antilogarithm is \(\frac{1}{6}\). This means we want to find a number 'x' such that \(6^{x} =\frac{1}{6}\)

Step 2: Express equation in exponential form

We know that \(\frac{1}{6} = 6^{-1}\). Therefore, the exponent 'x' equals -1.

Step 3: Conclude

From the previous step, we've found that 'x' is equal to -1. Therefore, \(\log _{6} \frac{1}{6} = -1\).