Generate Exponential Growth Formula Based on End Value

Question

As the title suggests I'm attempting to generate an exponential growth formula giving an initial, and final value. I am writing this for a videogame, where there are 8 skills that the player can level up. The main level of the character depends on a culmination of the other 8 skills. The specific formula I used was:

base * (level ^ slider)

Where base = 25, and Slider = 1.8.

Each skill can go up to level 100. With the formula in mind, each skill at level 100 requires a total of 3,604,395 exp. Since there are 8 skills, that means there is a total of 28,835,160 exp between all 8 levels. I need to write an exponential formula to end in 28,835,160. How would I go about doing this?

Answer 1

Note that an exponential function doesn't "end", but we can make it go through two specified points.

Let your exponential function be $$y=be^{ax}.$$

You have two points $(x_1,y_1)$ and $(x_2,y_2)$, which you know satisfy this equation.

Now simply insert the points and solve for $a$ and $b$:

$$y_1=be^{ax_1} \quad\text{and} \quad y_2=be^{ax_2} \implies$$ $$\frac{y_2}{y_1}=e^{a(x_2-x_1)} \implies$$ $$a =\frac{\ln\left(\frac{y_2}{y_1} \right)}{x_2-x_1} \quad\text{and} \quad b= \frac{y_1}{\left(\frac{y_2}{y_1}\right)^{x_1/(x_2-x_1)}}$$