Problem 107 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The evaluated result of the expression \(\log _{2}(\log _{3} 81)\) is 2.

Step 1: Understanding Logarithms

In this step, we recognize that a logarithm is an exponent. \(\log _{3} 81\) is equivalent to the question: which power must 3 be raised to, to obtain 81. Similarly, \(\log _{2}\) asks for the power for base 2.

Step 2: Evaluating the Inner Logarithm

Here, we evaluate \(\log _{3} 81\). It can be seen that \(3^4\) equals 81, so \(\log _{3} 81\) equals 4.

Step 3: Evaluating the Outer Logarithm

Substitute the result from Step 2 into the original expression. It becomes \(\log _{2} 4\). It can be seen that \(2^2\) equals 4, so \(\log _{2} 4\) equals 2.