[Math] If $\log_8 3 = P$ and $\log_3 5 = Q$, express $\log_{10} 5$ in terms of $P$ and $Q$.

Question

If $\log_8 3 = P$ and $\log_3 5 = Q$, express $\log_{10} 5$ in terms of $P$ and $Q$. Your answer should no longer include any logarithms.

I noted that $\log_5 10=\frac{1}{\log_{10} 5}.$

I also noted that $\log_{5} 10=\log_5 2+\log_5 5=\log_5 2+1.$ I don't know how to continue, how do I finish this problem using my strategy?

Answer 1

$$\log_83=P \iff \log_{2^3}3=P \iff \frac{1}{3}\log_23=P \iff \log_23=3P \iff \log_32=\frac{1}{3P}.$$

$$\log_35=Q.$$ Dividing last equalities at the end of both lines, we get: $$\log_52=\frac{1}{3PQ}$$ Also, we know $$\log_{10}5=\frac{1}{\log_510}=\frac{1}{\log_52\cdot5}=\frac{1}{\log_52+\log_55}=\frac{1}{\log_52+1}=\frac{1}{\frac{1}{3PQ}+1}=\frac{3PQ}{3PQ+1}$$