Problem 33 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The value of \(\log _{64} 8\) is \(0.5\)

Step 1: Identify Common Base

Rewrite \(64\) and \(8\) as \(2^6\) and \(2^3\) respectively, to identify a common base

Step 2: Apply Change of Base Formula

Apply change of base formula with base \(2\). This transforms \(\log _{64} 8 = \log _{2^6} 2^3\) into \(\frac {\log _2 2^3} {\log _2 2^6}\)

Step 3: Simplify Logarithm

Simplify \(\log _2 2^3\) as \(3\) and \(\log _2 2^6\) as \(6\)

Step 4: Conclude

After simplifying, you have \(\frac {3} {6}\) which equals \(0.5\)