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)
Answers
GPT-4o mini
The change of base formula for logarithms states that for any positive numbers \( a \), \( b \), and \( c \) (where \( a \neq 1 \) and \( b \neq 1 \)), the logarithm can be expressed as:
\[
\log_b a = \frac{\log_c a}{\log_c b}
\]
To evaluate \( \log_8 256 \) more easily, we can use base 2 since both 8 and 256 can be expressed as powers of 2. Therefore:
\[
\log_8 256 = \frac{\log_2 256}{\log_2 8}
\]
This makes it easier to calculate since \( \log_2 256 = 8 \) (because \( 256 = 2^8 \)) and \( \log_2 8 = 3 \) (because \( 8 = 2^3 \)), leading to:
\[
\log_8 256 = \frac{8}{3}
\]
\[
\log_b a = \frac{\log_c a}{\log_c b}
\]
To evaluate \( \log_8 256 \) more easily, we can use base 2 since both 8 and 256 can be expressed as powers of 2. Therefore:
\[
\log_8 256 = \frac{\log_2 256}{\log_2 8}
\]
This makes it easier to calculate since \( \log_2 256 = 8 \) (because \( 256 = 2^8 \)) and \( \log_2 8 = 3 \) (because \( 8 = 2^3 \)), leading to:
\[
\log_8 256 = \frac{8}{3}
\]