Question

log_{81}27

Explanation:

Step1: Express 81 and 27 as powers of 3

We know that \(81 = 3^4\) and \(27 = 3^3\). So we can rewrite the logarithm \(\log_{81}27\) as \(\log_{3^4}3^3\).

Step2: Apply the change of base formula for logarithms

The change of base formula for logarithms is \(\log_{a^m}a^n=\frac{n}{m}\) (derived from the property \(\log_{a^m}b=\frac{\log_ab}{\log_aa^m}=\frac{\log_ab}{m}\) and here \(b = a^n\)).
Using this property for \(\log_{3^4}3^3\), we get \(\frac{3}{4}\).

Answer:

\(\frac{3}{4}\)