Write $5 log_4(x) − log_\frac{1}{4}(y) + 4$ as a single logarithm

Question

$5\log_4(x) − \log_\frac{1}{4}(y) + 4$

The $4$ and $1/4$ are bases. I don't know how to simplify this if the bases are different.

Answer 1

You can change the base: \begin{equation} \log_{\frac{1}{4}}y=\frac{\log_{4}y}{\log_{4}\frac{1}{4}}=-\log_{4}y \end{equation}

Now \begin{equation} 5\log_4(x) − \log_\frac{1}{4}(y) + 4=5\log_4(x)+\log_4(y)+4=5\log_4(x)+\log_4(y)+4=\\ 5\log_4(x)+\log_4(y)+\log_4(4^4)=\log_4 (4^4x^5y) \end{equation}