Thanks to Kalid Azad's book (betterexplained.com), I understand exponential phenomena better.

ae^(rt) gives the growth of 'a' after 't' unit of times and with a "continuous" growth rate of 'r' (so if r*t = 1 like in e^1 so the formula will output the new value of 'a' after a 100% of growth during 1 unit of time 't', it can also thought as a 50% growth during 2 period of times etc...).

The ln(x) function outputs the amount of time needed to have a certain growth of the quantity '1'. e.g. ln(2.71...) = 1 (we need 1 unit of time to transition from 1 to 2.71... with 100% continuous growth).

My definitions are not 100% mathematical and precise but I can't visualize and understand exp(x) or ln(x) without them (especially their applications in engineering stuff).

The ln(x) function is the antiderivative of 1/x (or 1/x is the derivative of ln) and my question is what's the link between 1/x and the time needed to have a continuous growth of a rate "r" and during x unit of times. For example, why the derivative of the ln(x), the function returning the time to achieve 100% growth during a time unit, is the inverse of the time unit x. What's the intuitive explanation of ln(x) being the antiderivative of 1/x ?

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,...