# The number of users on a website has grown exponentially since its launch. After 2 months, there were 300 users. After 4 months there were 30000 users. Find the exponential function that models the number of users x months after the website was launched.

Exponential models
The number of users on a website has grown exponentially since its launch. After 2 months, there were 300 users. After 4 months there were 30000 users. Find the exponential function that models the number of users x months after the website was launched.

2021-02-22

Consider the model $$y=a \cdot b^{x}$$ which represents the exponential function that models the number of users x months after the website was launched.
Given that after 2 months, there were 300 users and after 4 months there were 30000 users.
This gives the equations
$$\displaystyle{300}={a}\cdot{b}^{{2}}$$ (1)
$$\displaystyle{30000}={a}\cdot{b}^{{4}}$$ (2)
Divide (2) by (1) and obtain the value of b as follows.
$$\displaystyle{\frac{{{a}\cdot{b}^{{4}}}}{{{a}\cdot{b}^{{2}}}}}={\frac{{{30000}}}{{{300}}}}$$
$$\displaystyle{b}^{{2}}={100}$$
$$\displaystyle{b}={10}$$
Substitute $$b=10$$ in (1) and simplify as follows.
$$\displaystyle{300}={a}\cdot{\left({10}\right)}^{{2}}$$
$$\displaystyle{300}={a}\cdot{100}$$
$$\displaystyle{a}={\frac{{{300}}}{{{100}}}}$$
$$\displaystyle{a}={3}$$
Thus, the exponential function that models the number of users x months after the website was launched is $$\displaystyle{y}={3}\cdot{\left({10}\right)}^{{x}}$$