\[log_{11} x +log_6 y\]
, we have to change one of the bases so they are the same.We can change the 11 into an 6,by solvimg
\[11=6^a \]
, taking the logarithms of both sides to give \[log 11 =a log 6 \rightarrow a=\frac{log 11}{log 6}\]
.Then
\[log_{6^{\frac{log 11}{log 6}} x +log_6 y\]
.Use the change of base rule to give
\[\frac{log 6}{log 11}log_6 x +log_6 y\]
Take the coefficient inside the logarithm.
\[ log_6 x^{\frac{log 6}{log 11}} +log_6 y\]
Hence
\[log_6 (x^{\frac{log 6}{log 11}} y)\]