Question

How to find the equation of an exponential function given two points?

Answer 1

An exponential function is given by $y=a(b)^x$

Given two points, (m, n) and (p, q), we find the equation of the exponential function as follows:

$n=a(b)^m \ .\ .\ .\ (1) \\ \\ q=a(b)^p \ .\ .\ .\ (2) \\ \\ \frac{(1)}{(2)} \Rightarrow \frac{n}{q} = \frac{b^m}{b^p} =b^{m-p} \\ \\ \Rightarrow \ln\left( \frac{n}{q} \right)=(m-p)\ln(b) \\ \\ \Rightarrow \ln(b)= \frac{\ln\left( \frac{n}{q} \right)}{m-p} \\ \\ \Rightarrow b=e^{\frac{\ln\left( \frac{n}{q} \right)}{m-p}$

From (1), we have:

$n=a\left(e^{m\frac{\ln\left( \frac{n}{q} \right)}{m-p}\right) \\ \\ \Rightarrow a= \frac{n}{\left(e^{m\frac{\ln\left( \frac{n}{q} \right)}{m-p}\right)}}$

Therefore, the equation of an exponential function given two points (m, n) and (p, q) is given by

$y=\frac{n}{\left(e^{m\frac{\ln\left( \frac{n}{q} \right)}{m-p}\right)}}\left(e^{\frac{\ln\left( \frac{n}{q} \right)}{m-p}\right)^x$

[i.e. you can choose any set of points and substitute the values in the equation above to get the exponential equation]