Question
Recall what you know about the change of base formula. Using the formula, write a new logarithm in 1–2 sentences so that log_8 256
can be easily evaluated.
can be easily evaluated.
Answers
GPT-4o mini
The change of base formula for logarithms states that \(\log_b a = \frac{\log_k a}{\log_k b}\) for any positive base \(k\). To evaluate \(\log_8 256\) more easily, we can express it using base 2:
\[
\log_8 256 = \frac{\log_2 256}{\log_2 8}
\]
Since \(256 = 2^8\) and \(8 = 2^3\), we can further simplify as \(\log_2 256 = 8\) and \(\log_2 8 = 3\), allowing us to compute \(\log_8 256 = \frac{8}{3}\).
\[
\log_8 256 = \frac{\log_2 256}{\log_2 8}
\]
Since \(256 = 2^8\) and \(8 = 2^3\), we can further simplify as \(\log_2 256 = 8\) and \(\log_2 8 = 3\), allowing us to compute \(\log_8 256 = \frac{8}{3}\).