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
can be easily evaluated
Answers
GPT-4o mini
The change of base formula for logarithms states that for any positive numbers \( a \), \( b \), and \( c \) (where \( a \) and \( b \) are not equal to 1), the logarithm can be expressed as:
\[
\log_b a = \frac{\log_c a}{\log_c b}
\]
To evaluate \( \log_8 256 \) easily, we can use base 2, giving us:
\[
\log_8 256 = \frac{\log_2 256}{\log_2 8}
\]
Now, since \( 256 = 2^8 \) and \( 8 = 2^3 \), we can calculate:
\[
\log_2 256 = 8 \quad \text{and} \quad \log_2 8 = 3
\]
Thus,
\[
\log_8 256 = \frac{8}{3} = \frac{8}{3}.
\]
\[
\log_b a = \frac{\log_c a}{\log_c b}
\]
To evaluate \( \log_8 256 \) easily, we can use base 2, giving us:
\[
\log_8 256 = \frac{\log_2 256}{\log_2 8}
\]
Now, since \( 256 = 2^8 \) and \( 8 = 2^3 \), we can calculate:
\[
\log_2 256 = 8 \quad \text{and} \quad \log_2 8 = 3
\]
Thus,
\[
\log_8 256 = \frac{8}{3} = \frac{8}{3}.
\]