I am not very experienced in mathematical notation, so please excuse some terminology misuse or formatting shortcomings.
I have a project in which a value needs to increase from a set minimum to a set maximum in a set number of seconds. It is easy to calculate the linear value based on ratios.
Let $v$ = desired value, $n$ = minimum limit, $x$ = maximum limit, $t$ = elapsed time, and $t_x$ = allocated time:
$v = \frac{t}{t_x}(x-n) + n.$
Thus if my values are:
$n = 5, x = 90, t_x = 1800 \text{ (half hour)}$
For elapsed time of $5$ minutes ($600$ s):
$v = \frac{600}{1800} (90-5) + 5 = 33.3.$
The problem is I want to change this linear growth to exponential growth, and I'm not sure how to alter the formula.
So instead of $33.3$ at $5$ minutes, I would rather have $13$ for example. (Slow initial change, rapid later change.)
How can I insert an exponential growth factor into my equation and honor the minimum and maximum values allowed?