Question

Rewrite the logarithm log
9
​
81 using the change of base formula with a base of 3, then evaluate.
log
9
​
81=□

Answer 1

To rewrite \(\log_9 81\) using the change of base formula with a base of 3, we use the formula:

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

In this case, we will set \(a = 9\), \(b = 81\), and \(c = 3\). Thus, we have:

\[
\log_9 81 = \frac{\log_3 81}{\log_3 9}
\]

Next, we evaluate \(\log_3 81\) and \(\log_3 9\):

1. \(81\) can be expressed as \(3^4\):
\[
\log_3 81 = \log_3 (3^4) = 4
\]

2. \(9\) can be expressed as \(3^2\):
\[
\log_3 9 = \log_3 (3^2) = 2
\]

Now we substitute these values back into the change of base formula:

\[
\log_9 81 = \frac{\log_3 81}{\log_3 9} = \frac{4}{2} = 2
\]

Thus, the final result is:

\[
\log_9 81 = 2
\]