How far ? (Inspired by Sir Pi Han Goh)

$\begin{aligned} 2^2 & = 4 \\ 3^3 & = 27 \\ 4^4 & = 256 \\ 5^5 & = 3125 \\ \vdots \ & = \quad \vdots \end{aligned}$

$N^N$ has exactly $N -1$ digits in the right-hand side of above some illustrations.

True or False?

Does $N^N$ has exactly $N-1$ digits for $N> 1$ and $N$ is not equal to multiple integral of $10$ ?