Thanks to Kalid Azad's book (betterexplained.com), I understand exponential phenomena better.
ae^(rt)$ae^{rt}$ gives the growth of 'a''$a$' after 't''$t$' unit of times and with a "continuous" growth rate of 'r''$r$' (so if r*t = 1$rt = 1$ like in e^1 so the$e^1$ the formula will output the new value of 'a''$a$' after a 100%$100\%$ of growth during 1$1$ unit of time 't''$t$', it can also be thought of as a 50%$50\%$ growth during 2 period$2$ periods of times, etc...).
The ln(x)$\ln(x)$ function outputs the amount of time needed to have a certain growth of the quantity '1''$1$'. e.g. ln(2.71...) = 1 $\ln(2.71\dots) = 1$ (we need 1$1$ unit of time to transition from 1$1$ to 2.71...$2.71\dots$ with 100%$100\%$ continuous growth).
My definitions are not 100%$100\%$ mathematical and precise but I can't visualize and understand exp(x)$\exp(x)$ or ln(x)$\ln(x)$ without them (especially their applications in engineering stuff).
The ln(x)$\ln(x)$ function is the antiderivative of 1/x$\frac1x$ (or 1/x$\frac1x$ is the derivative of ln$\ln(x)$), and my question is what's the link between 1/x$\frac1x$ and the time needed to have a continuous growth of a rate "r""$r$" and during x$x$ unit of times. For example, why the derivative of the ln(x)$\ln(x)$, the function returning the time to achieve 100%$100\%$ growth during a time unit, is the inverse of the time unit x$x$. What's the intuitive explanation of ln(x)$\ln(x)$ being the antiderivative of 1/x$\frac1x$ ?
e^x = e^1, e^2, e^3, e^4,... ln(e^x) = 1, 2, 3, 4,... 1/x = 1/e^1, 1/e^2, 1/e^3, 1/e^4,...$$e^x = e^1, e^2, e^3, e^4,\dots$$
$$\ln(e^x) = 1, 2, 3, 4,\dots$$
$$\frac1x = \frac1{e^1}, \frac1{e^2}, \frac1{e^3}, \frac1{e^4},\dots$$
