4B. For which natural number $n$ does the expression $\frac{\lg 2 \cdot \lg 3 \cdot \lg 4 \ldots \ldots \lg n}{10^{n}}$ attain its smallest value?
Solution. Let $\frac{\lg 2 \cdot \lg 3 \cdot \lg 4 \ldots \cdot \cdot \lg n}{10^{n}}=a_{n}$. Then $a_{n}=a_{n-1} \frac{\lg n}{10}$. From this, it is clear that:

- if $\frac{\lg n}{10}<1$, then $a_{n}<a_{n-1}$;
- if $\frac{\lg n}{10}=1$, then $a_{n}=a_{n-1}$;
- if $\frac{\lg n}{10}>1$, then $a_{n}>a_{n-1}$.

Equality is satisfied only for $n=10^{10}$; for $n<10^{10}$, $a_{n}<a_{n-1}$, and for $n>10^{10}$, $a_{n}>a_{n-1}$.

Thus, the sequence $a_{1}, a_{2}, \ldots$ decreases up to $n=10^{10}-1$, for $n=10^{10}-1$ and $n=10^{10}$ the same terms are obtained, and then the sequence increases. The smallest value of $a_{n}$ is therefore obtained for $n=10^{10}$ and $n=10^{10}-1$.