Question

Given log8 32 , change the base to an integer other than 10 to easily evaluate the logarithm.(1 point)

Answer 1

To evaluate \( \log_8 32 \) using a different base, we can use the change of base formula:

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

where \( c \) is any positive integer base. A common and convenient choice for many logarithmic evaluations is base \( 2 \).

Using base \( 2 \):

\[
\log_8 32 = \frac{\log_2 32}{\log_2 8}
\]

Now we need to calculate each logarithm separately:

1. **Calculate \( \log_2 32 \)**:
\[
32 = 2^5 \quad \text{therefore} \quad \log_2 32 = 5
\]

2. **Calculate \( \log_2 8 \)**:
\[
8 = 2^3 \quad \text{therefore} \quad \log_2 8 = 3
\]

Now we can substitute these values back into our equation:

\[
\log_8 32 = \frac{\log_2 32}{\log_2 8} = \frac{5}{3}
\]

Thus, \( \log_8 32 = \frac{5}{3} \).