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 \(\displaystyle{y}={3}\cdot{\left({10}\right)}^{{x}}\)