I have two points:
$$A(X_0,Y_0) $$ $$ B(X_1,Y_1)$$
And I need to find the function that creates an exponential growth between the two point. The fuction for exponential growth is $y = ab^x$ and then
$$y_0=a\cdot b^{x_0}$$ $$y_1=a\cdot b^{x_1}$$
and solve for $a$
$$a= \frac{b^{x_0}}{y_0}$$ $$a= \frac{b^{x_1}}{y_1}$$
because we want the same curve for the two equations we can say that they are equivalent:
$$\frac{b^{x_0}}{y_0} = \frac{b^{x_1}}{y_1}$$
or
$$y_1 \cdot b^{x_0} = y_0 \cdot b^{x_1}$$
and then
$$b^{x_1-x_0} = \frac{y_1}{y_0}$$ $$y_1 - b^{x_1-x_0} \cdot y_0= 0$$
Now let's plug in our a equation our solution:
$$y_1 - b^{x_1-x_0} \cdot y_0 = \frac{b^{x_0}}{y_0}$$
or
$$y_0\cdot y_1-b^{x_1-x_0}-b^{x_0} = 0$$
Now I'm stuck here because I have a point $C$ with $y_0 < Y_c > y_1$ and I need to know the $X_c$ that falls on the curve we have calculated.
Just divide the equation $y_0=ab^{x_0}$ by the equation $y_1=ab^{x_1}$ to find $b$. Then plug in the coordinates of either point to compute $a$.