Problem 106 - College Algebra Essentials

Question

Evaluate each expression without using a calculator. $$\log _{5}\left(\log _{2} 32\right)$$

Step-by-Step Solution

Answer

The value of the expression \(\log _{5}\left(\log _{2} 32\right)\) is 1.

Step 1: Evaluate The Inner Logarithm

Begin by evaluating the inner logarithm \(\log_2 32\). Given that \(2^5 = 32\), it follows that \(\log_2 32 = 5\). So the expression simplifies to \(\log_5 5\).

Step 2: Evaluate The Outer Logarithm

The next step is to evaluate \(\log_5 5\). Given that \(5^1 = 5\), it is clear that \(\log_5 5 = 1\). Therefore, the expression simplifies to 1.