An Intution for Exponentials

One way to see exponentials is as numbers decomposed into “infinite roots”. Look at Bernoulli’s original formulation of $e$:

By abusing notation and using $dx$ to mean the equation’s infitesimally small number, we can rewrite as

which is $e$’s decomposition into its infinite root $1+dx$, and it’s easy to see why the derivative of $e^x$ is $e^x$:

Other (non-negative) numbers can be similiarly decomposed:

The last step follows from the substitution of $n\ln a$ for $n$ in Bernoulli’s formula above. Thus, a base $a$ number grows at a rate $\ln a$ times $e$’s unit growth rate.