AHSME 1974 Problem 18

If $\log_8{3}=p$ and $\log_3{5}=q$, then, in terms of $p$ and $q$, $\log_{10}{5}$ equals $\mathrm{(A)\ } pq \mathrm{(B) \ }\frac{3p+q}{5} \mathrm{(C) \ } \frac{1+3pq}{p+q} \mathrm{(D) \ } \frac{3pq}{1+3pq} \mathrm{(E) \ }p^2+q^2$