Question

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

Answer 1

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

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

where \( c \) is the new base. A convenient choice for the new base could be 2, since both 8 and 32 are powers of 2.

First, we can express 8 and 32 as powers of 2:
- \( 8 = 2^3 \)
- \( 32 = 2^5 \)

Now we can rewrite \( \log_8 32 \) using base 2:

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

Evaluating each logarithm:
- \( \log_2 32 = 5 \) (since \( 32 = 2^5 \))
- \( \log_2 8 = 3 \) (since \( 8 = 2^3 \))

Now substituting these values into the formula gives us:

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

Therefore, the final result is:

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