Simplification of an exponential equation.

Question

I am currently in a class on sinusoids and exponential functions (that is confusing the hell out of me...).

My professor has the follow equation (I believe it is the general equaltion for exponential growth but he defines everything based on t (time)...)

This is the equation he gives in the pdf

s(t) = A * e^a t Note: A != a

then he says lets introduce another variable T to make the equation more 'easy' to understand Note: T != t :-/

T = 1/a This being the unit of time in which the function s(t) increases by e.

He then pulls out this magical equation

s(t) = A * e ^ (t/T)

I have gotten to the following from the original equation

s(t) = A * t/(e^T)

How do I get to what my professor has?

Answer 1

As has been established in the comments below, the expression $Ae^at$ is a typo. Note, this is a very common typo and arises in $\LaTeX$ when there is more than one letter in the exponent (or more than one letter in an index). For example Ae^at produces $Ae^at$; what is required to obtain the desired result is braces: Ae^{at} which produces $Ae^{at}$.

---

Note, $T$ is a constant, not a variable. In addition, $T$ is the amount of time it takes for $s(t)$ to increase by a factor of $e$ which means $s(t+T) = es(t)$. What you wrote in your post would mean $s(t+T) = s(t) + e$ which is false.

As $T = \dfrac{1}{a}$, we have $a = \dfrac{1}{T}$. So the equation becomes $$s(t) = Ae^{at} = A\exp(at) = A\exp\left(\frac{1}{T}t\right) = A\exp\left(\frac{t}{T}\right).$$