Expand the Logarithmic Expression log base 8 of 1/64

Problem

(log_8)(1/64)

Solution

1. Identify the base and the argument of the logarithm. The base is 8 and the argument is the fraction 1/64

2. Apply the quotient rule for logarithms, which states that (log_b)(x/y)=(log_b)(x)−(log_b)(y)

(log_8)(1/64)=(log_8)(1)−(log_8)(64)

3. Evaluate the first term. Since any base raised to the power of 0 is 1 the logarithm of 1 is always 0

(log_8)(1)=0

4. Evaluate the second term by expressing 64 as a power of the base 8 Since 8=64 the logarithm is 2

(log_8)(64)=2

5. Subtract the values to find the final expanded result.

0−2=−2

Final Answer

(log_8)(1/64)=−2

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