/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 78 Evaluate each expression without... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 78

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

Short Answer

Expert verified
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Step by step solution

01

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, \( \log _{2} 32 = 5 \).
02

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, \( \log _{5} 5 = 1 \).
03

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.

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