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