Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$?

Question

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$? There were some other questions like this and I applied this formula to them $\log_a(xy) = \log_a(x)+\log_a(y)$. However, in this question, I can't because I can't evaluate $\log_x(y)$. What do I do?

Answer 1

$\log_x{4y} = \log_x{4} + \log_x{y}$

We know that $\log_x{a} = \frac{\log_4{a}}{\log{x}}$, so $\log_x{4y} = \frac{\log_4{4} + \log_4{y}}{\log_4{x}} = \frac{q+1}{\log_4{x}}$

We also know $\log_2{x} = p$, so $2^p = x$

and

$\left(4^{\frac{1}{2}}\right)^p = x$

and

$4^{\frac{p}{2}} = x$

So $\log_4{x} = \frac{p}{2}$, and we're done.

$\log_x{4y} = \frac{2(q+1)}{p}$

Answer 2

$$\log_{x}(4y)=\frac{\ln 4+\ln y}{\ln x}=\frac{\ln 4+q\ln 4}{p\ln 2}=\frac{2\ln 2(q+1)}{p\ln 2}=\frac{2(q+1)}{p}$$