Problem 79 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$\log _{2}\left(\log _{3} 81\right)$$

Step-by-Step Solution

Answer

The answer is 2

Step 1: Evaluate inner logarithm

Begin by evaluating the logarithm \(\log _{3} 81\). This logarithm asks for which exponent would be necessary to raise 3 to get 81. Since \(3^4 = 81\), it means the value of \(\log _{3} 81\) is 4.

Step 2: Substitute for outer logarithm

Substitute the value from the inner logarithm into the outer. It changes \(\log _{2}\left(\log _{3} 81\right)\) to \(\log _{2}(4)\). Again, this is asking for the exponent needed to raise 2 to get 4. We know that \(2^2 = 4\), so \(\log _{2}(4)\) equals to 2.