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Chapter 4: Problem 107

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.

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