I am trying to derive the formula for the fundamental period of an exponential in the form $e^{j\omega t}$ where $j$ is an imaginary number and $w\omega$ is the frequency.
If $e^{j\omega t}$ is periodic, then:
$$e^{j\omega t} = e^{j\omega(t+T)} = e^{j\omega t}\cdot e^{j\omega t}$$
must be true for some period $T$.
Thus:
$$e^{j\omega T} = 1$$
I tried to take $\ln$ of both sides, but this failed to give me the equation for the fundamental period in the textbook ($T = 2\pi/|\omega|$).
If $e^{i \theta}=1$ then $\theta = 2 \pi n$ where $n$ is an integer. For your variables, this means $\omega T=2 \pi n$. For 1 period, $(n=1)$ we then have $T=2 \pi/\omega$