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)

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?

Michael Albanese · Accepted Answer · 2013-09-22T23:44:16.460

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 $$s(t) = Ae^{at} = A\exp(at) = A\exp\left(\frac{1}{T}t\right) = A\exp\left(\frac{t}{T}\right).$$

edited Sep 22 '13 at 23:44

answered Sep 22 '13 at 23:25

Michael Albanese

99,526

s(t) = A * (e^a) * t; t isn't raising e to the power of anything. – Zimm3r Sep 22 '13 at 23:27
If that is the case, you do not have an exponential function but a linear function. Are you sure the $t$ isn't in the exponent? It might be a typo. – Michael Albanese Sep 22 '13 at 23:29
I uploaded a picture of the pdf – Zimm3r Sep 22 '13 at 23:30
If the section/chapter/pdf title is about exponential functions, then I believe this is a typo. – Michael Albanese Sep 22 '13 at 23:31
Ok I looked later down where he writes it for the decaying functions and writes s(t) = A*e^(−αt) – Zimm3r Sep 22 '13 at 23:32
It is definitely a typo, in which case what I wrote works. – Michael Albanese Sep 22 '13 at 23:33
Ok. Thanks I was getting even more confused haha. – Zimm3r Sep 22 '13 at 23:34

Simplification of an exponential equation.

1 Answers1