Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$\log _{5} \frac{1}{5}$$

Step-by-Step Solution

Answer

The value of \(\log _{5} \frac{1}{5}\) is -1

Step 1: Understand the logarithm definition

Based on the definition of a logarithm, \(\log _{b} a=n\) if and only if \(b^{n}=a\). It implies that the base \(b\), raised to the power \(n\), equals \(a\).

Step 2: Apply the logarithm definition to the given expression

Applying the definition to the given expression would mean finding a number 'n' such that when 5 (the base) is raised to the power 'n' equals \(\frac{1}{5}\). This can be written as \(5^n = \frac{1}{5}\)

Step 3: Transform the expression

It is evident that \( \frac{1}{5}\) can be written as \(5^{-1}\), therefore, \(5^n = 5^{-1}\).

Step 4: Compare the exponentials

Using the property of equality of expressions, n should be equal to -1. This is because the base of both sides is the same (which is 5), thus their exponents must equal each other. Hence \(n = -1\)