Replacing "e" in Euler's formula with another number
Say you're interested in $$M^{j2\pi f_0 t}. \tag{1}$$ Note that $$M = e^{\log M},$$ so $(1)$ can be written as

\begin{align}
M^{j2\pi f_0 t} &amp;= \left( e^{\log M} \right) ^ {j2\pi f_0 t} \\
&amp;= e^{j2\pi (f_0\log M) t} \\
&amp;= \cos(2\pi (f_0\log M) t) + j \sin(2\pi (f_0\log M) t),
\end{align}
which is a complex sinusoid with frequency $f_0 \log M$. That is why using $M$ instead of $e$ results in a change in frequency.