Problem 78 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

1

Step 1: Evaluate Inner Logarithm

First, evaluate the inner expression \( \log _{2} 32 \). By definition of logarithm, we are trying find the exponent \( x \) that satisfies \( 2^x = 32 \). By simple exponential operations, we find that \( x = 5 \). Thus, \( \log _{2} 32 = 5 \).

Step 2: Evaluate Outer Logarithm

Next, substitute the result from step 1 into the outer expression \( \log _{5} \). So, we get \( \log _{5} 5 \), which means we try to find the exponent \( y \) that satisfies \( 5^y = 5 \). Again, by simple exponential operations, we see that \( y = 1 \). Therefore, \( \log _{5} 5 = 1 \).

Step 3: Write Final Answer

The resulting answer from step 2 is the final answer, hence the solution to the provided expression \( \log _{5}\left(\log _{2} 32\right) \) is 1.