Question
Recall what you know about the change of base formula. Using the formula, write a new logarithm in 1–2 sentences so that log8256
can be easily evaluated.(2 points)
can be easily evaluated.(2 points)
Answers
GPT-4o mini
The change of base formula allows us to convert a logarithm to a different base, which can simplify evaluation. For \( \log_8 256 \), we can express it using base 2:
\[
\log_8 256 = \frac{\log_2 256}{\log_2 8}.
\]
Since \( 256 = 2^8 \) and \( 8 = 2^3 \), this can be calculated as:
\[
\log_2 256 = 8 \quad \text{and} \quad \log_2 8 = 3,
\]
thus,
\[
\log_8 256 = \frac{8}{3}.
\]
\[
\log_8 256 = \frac{\log_2 256}{\log_2 8}.
\]
Since \( 256 = 2^8 \) and \( 8 = 2^3 \), this can be calculated as:
\[
\log_2 256 = 8 \quad \text{and} \quad \log_2 8 = 3,
\]
thus,
\[
\log_8 256 = \frac{8}{3}.
\]