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Chapter 4: Problem 33

Evaluate each expression without using a calculator. $$\log _{64} 8$$

Short Answer

Expert verified
The value of \( \log_{64} 8 \) is 0.5

Step by step solution

01

Understand the logarithm equation

Express the logarithm equation \( \log_{64} 8 \) as the exponential equation \( 64^x = 8 \)
02

Express the numbers with the same base

Most notably, it is possible to express both 64 and 8 as powers of 2: \( 64 = 2^6 \) and \( 8 = 2^3 \). This allows us to represent the exponential equation from Step 1 with a common base: \( (2^6)^x = 2^3 \)
03

Simplify the exponential equation

Simplifying the left side yields: \( 2^{6x} = 2^3 \). We can reason that if \( 2^{6x} = 2^3 \), then the exponents must be equal, so \( 6x = 3 \)
04

Solve for x

Solving for x, we get \( x = \frac{3}{6} = 0.5 \)

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