Question 156
For what positive integers $n \gt 1$ does the expression
$$
\frac{\log_{10} 2 \cdot \log_{10} 3 \cdots \log_{10} n}{10^{n-1}}
$$
attain the smallest possible value? Find this value.
$$
\frac{\log_{10} 2 \cdot \log_{10} 3 \cdots \log_{10} n}{10^{n-1}}
$$
attain the smallest possible value? Find this value.