I was working on this Problem Prove that: $$ \frac{\log_5(nt)^2}{\log_4\left(\frac{t}{r}\right)}=\frac{2\log(n)\cdot\log(4)+2\log(t)\cdot \log(4)}{\log(5)\cdot \log(t)-\log(5)\cdot\log(r)}$$
I think it has something to do with change of base because it's $\log_{10}$ on the right side and not on the left, but I'm not sure how to go about this.
\begin{align} \frac{\log_5(nt)^2}{\log_4(t/r)} &= \frac{\frac{\log (nt)^2 }{\log 5}}{\frac{\log(t/r)}{\log 4}} \\ &= \frac{\log 4 \log(nt)^2}{\log 5 \log(t/r)} \\ &= \frac{2\log 4 (\log(n)+\log(t))}{\log 5 (\log(t)-\log (r))} \end{align}
where the following identities are used:
$$\log_a b = \frac{\log_c b}{\log_c a}$$ in the first equality.
Also $$\log a^n= n\log a,$$ $$\log(ab)=\log a + \log b,$$ and $$\log(a/b)=\log a - \log b$$