Question

Use the change of base formula to rewrite the logarithm: log base 8 256

Answer 1

The change of base formula for logarithms states that

\[
\log_b a = \frac{\log_c a}{\log_c b}
\]

for any positive number \( c \).

In this case, we want to calculate \( \log_8 256 \). We can use the natural logarithm (base \( e \)) or the common logarithm (base \( 10 \)). Here, I will use the common logarithm (base \( 10 \)):

\[
\log_8 256 = \frac{\log_{10} 256}{\log_{10} 8}
\]

To calculate \( \log_{10} 256 \) and \( \log_{10} 8 \):

1. We can express \( 256 \) as \( 2^8 \) and \( 8 \) as \( 2^3 \):
- Hence, \( \log_{10} 256 = \log_{10} (2^8) = 8 \cdot \log_{10} 2 \)
- And \( \log_{10} 8 = \log_{10} (2^3) = 3 \cdot \log_{10} 2 \)

2. Substituting these into our formula gives:
\[
\log_8 256 = \frac{8 \cdot \log_{10} 2}{3 \cdot \log_{10} 2}
\]

3. The \( \log_{10} 2 \) terms cancel out:
\[
\log_8 256 = \frac{8}{3}
\]

Thus, the final answer is:

\[
\log_8 256 = \frac{8}{3}
\]