periodic in $x$ with period $2\pi$

67 Views Asked by At

What it means for a function to be

periodic in $x$ with period $\frac{2\pi}{\log(2)}$ ?

When I plug in $2^x$ in Wolfram Alpha I get this information.

Does it have something to do with the Euler number $e$ or $\pi$, derivation or integrals? Because Ive seen some constant number raised to some radi which represented angles, but i forgot about it. Because it feels periodicity has something to do with angles of some rotation in it? But it can also be a loop? or am I on a blueberry trip.

I also wonder if some function is periodic, does it link to closed-form expression or can they not be connected.

I read that

a periodic function is a function that repeats its values in regular intervals or periods

but how can $2^x$ repeat itself if $x$ is increasing sequentially? Is it just that $x$ on the Reals can be periodic $x$ is a sine function?