Problem 33
Question
Evaluate each expression without using a calculator. $$\log _{64} 8$$
Step-by-Step Solution
Verified Answer
The value of \( \log_{64} 8\) is \(\frac{1}{2}\)
1Step 1: Simplify the Bases
First, simplify the bases of 64 and 8 as they are both powers of 2. This transforms the expression into: \(\log_{2^6}(2^3)\)
2Step 2: Simplify Logarithm
Now apply the power rule of logarithms which brings down the exponent to the front. The expression becomes: \(\frac{1}{6} \log_{2}(2^3)\)
3Step 3: Simplify the Logarithm to Its Base
The base of the log and the number under the log are the same (both 2), so the expression simplifies to: \(\frac{1}{6} * 3 = \frac{1}{2}\)
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