Question

question 5 of 10
evaluate the following expression.
\log_{8} 64

a. 0.125
b. 0.5
c. 2
d. 8

Explanation:

Step1: Recall logarithm definition

Let \( \log_{a}b = x \), then \( a^{x}=b \). For \( \log_{8}64 \), let \( \log_{8}64 = x \), so \( 8^{x}=64 \).

Step2: Express 8 and 64 as powers of 2

We know \( 8 = 2^{3} \) and \( 64 = 2^{6} \). Substitute into the equation: \( (2^{3})^{x}=2^{6} \).

Step3: Simplify the left - hand side

Using the power - of - a - power rule \( (a^{m})^{n}=a^{mn} \), we get \( 2^{3x}=2^{6} \).

Step4: Solve for x

Since the bases are the same, we can set the exponents equal: \( 3x = 6 \). Divide both sides by 3: \( x=\frac{6}{3}=2 \).

Answer:

C. 2