The First Digit

Any positive number can be rewritten as $x=y \times 10^n$ , where $y$ is a real number with $1 \leq y < 10$ and $n$ is an integer.

Thus, $log_{10}(x) = n+\log_{10}y$

The probability of first digit of $x$ being 1 = The probability of first digit of $x$ being 1 = probability of $\log_{10}(1) \leq \log_{10}(y) < \log_{10}2$

The required probability is thus $p=\log_{10}{2}-log_{10}(1)=\boxed{0.301}$

This counter-intuitive property is confirmed by the Benford's Law