6・33 If $\log _{8} 3=p, \log _{3} 5=q$, then $\lg 5$ expressed in terms of $p$ and $q$ is
(A) $p q$.
(B) $\frac{3 p+q}{5}$.
(C) $\frac{1+3 p q}{p+q}$.
(D) $\frac{3 p q}{1+3 p q}$.
(E) $p^{2}+q^{2}$.
(25th American High School Mathematics Examination, 1974)
［Solution］ From the given, we have $3=8^{p}=2^{3 p}, 5=3^{q}$.
Therefore,
$$
5=\left(2^{3 p}\right)^{q}=2^{3 p q} .
$$

Thus $\lg 5=\lg 2^{3 p q}=3 p q \lg 2=3 p q \lg \frac{10}{5}$.
Since $\lg \frac{10}{5}=\lg 10-\lg 5=1-\lg 5$, we have
$$
\lg 5=3 p q(1-\lg 5),
$$

which simplifies to
$$
\lg 5=\frac{3 p q}{1+3 p q}
$$

Therefore, the answer is $(D)$.