Problem 26 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The evaluation of \(\log _{6} \frac{1}{6}\) is -1.

Step 1: Understand the problem

We wish to evaluate the expression \(\log _{6} \frac{1}{6}\). This is equivalent to finding a power to which 6 must be raised to give us the number \(\frac{1}{6}\).\The decimal result is not required in this question. Instead, an exact value in integer or fractional form is needed.

Step 2: Apply Logarithm Rule

Any positive number raised to the power -1 results in its reciprocal. Therefore, based on this fact, 6 raised to the power of -1 gives us \(\frac{1}{6}\). So the power to which 6 must be raised to get \(\frac{1}{6}\) is -1.

Step 3: Final Answer

Therefore, the expression \(\log _{6} \frac{1}{6}\) equals to -1.