Problem 25 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The evaluation of the expression \(\log_{5} \frac{1}{5}\) without using a calculator is -1.

Step 1: Rewriting the fraction

We rewrite the fraction as \(5^{-1}\) because \(1/5\) is equivalent to \(5^{-1}\). This will help us express the logarithm in a way in which we can solve it easily. So, we now have \(\log_{5}5^{-1}\).

Step 2: Applying Logarithmic Rule

Next, apply the rule of logarithms that states: \(\log_{b}a^n = n \cdot \log_{b}a\). Therefore, \(\log_{5}5^{-1} = -1 \cdot \log_{5}5\).

Step 3: Simplifying

\(\log_{b}b = 1\) for any number \(b\). So, we have \(-1 \cdot 1 = -1\).