3. For which natural numbers $n>1$ does the expression

$$
\frac{\log _{10} 2 \cdot \log _{10} 3 \cdots \log _{10} n}{10^{n-1}}
$$

attain its minimum value? What is this value?
III/3. Let's compare the expressions $\frac{\log _{10} 2 \log _{10} 3 \cdots \log _{10}(n-1)}{10^{n-2}}$ and $\frac{\log _{10} 2 \log _{10} 3 \cdots \log _{10} n}{10^{n-1}}$. The inequality

$$
\frac{\log _{10} 2 \log _{10} 3 \cdots \log _{10}(n-1)}{10^{n-2}} \geq \frac{\log _{10} 2 \log _{10} 3 \cdots \log _{10} n}{10^{n-1}}
$$

holds exactly when $1 \geq \frac{1}{10} \log _{10} n=\log _{10} \sqrt[10]{n}$, which is equivalent to $10 \geq \sqrt[10]{n}$ or $10^{10} \geq n$. From this, it follows that

$$
\begin{gathered}
\frac{\log _{10} 2}{10}>\frac{\log _{10} 2 \cdot \log _{10} 3}{10^{2}}>\ldots>\frac{\log _{10} 2 \cdot \log _{10} 3 \cdots \log _{10}\left(10^{10}-1\right)}{10^{10^{10}-2}} \\
=\frac{\log _{10} 2 \cdot \log _{10} 3 \cdots \log \left(10^{10}\right)}{10^{10^{10}-1}}
\end{gathered}
$$

and

$$
\begin{gathered}
\frac{\log _{10} 2 \cdot \log _{10} 3 \cdots \log \left(10^{10}\right)}{10^{10^{10}-1}}<\frac{\log _{10} 2 \cdot \log _{10} 3 \cdots \log \left(10^{10}+1\right)}{10^{10^{10}}} \\
\quad<\frac{\log _{10} 2 \cdot \log _{10} 3 \cdots \log \left(10^{10}+2\right)}{10^{10^{10}+1}}<\ldots
\end{gathered}
$$

Thus, the value of the expression $\frac{\log _{10} 2 \log _{10} 3 \cdots \log _{10} n}{10^{n-1}}$ is smallest for $n=10^{10}-1$ and $n=10^{10}$. It is equal to

$$
\frac{\log _{10} 2 \log _{10} 3 \cdots \log _{10} 10^{10}}{10^{10^{10}-1}}
$$

Comparison of expressions for $n$ and $n+1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ points

(1 point is awarded for comparing expressions only for small $n$.)

![](https://cdn.mathpix.com/cropped/2024_06_07_16ef8862fb9f11daffb9g-11.jpg?height=57&width=1627&top_left_y=1502&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_16ef8862fb9f11daffb9g-11.jpg?height=52&width=1633&top_left_y=1553&top_left_x=220)

The expression is smallest for $n=10^{10}-1$ and $n=10^{10}$

1 point