Given points on the exponential model: (0,1) and (5,16).

General equation of an exponential model: y=aebx

Evaluate the general equation at x=0 and x=5:

\(\displaystyle{y}={a}{e}^{{b}}{\left({0}\right)}={a}{e}^{{0}}={a}{\left({1}\right)}={a}\)

\(\displaystyle{y}={a}{e}^{{b}}{\left({5}\right)}={a}{e}^{{5}}{b}\)

We require y=1 when x=0 and y=16 when x=5 (as the points (0,1) and (5,16) need to lie on the exponential model).

a=1

\(\displaystyle{a}{e}^{{5}}{b}={16}\)

By the first equation, we thus know that a=1. Let us replace \(\displaystyle{a}{e}^{{5}}{b}={16}\) by 1 in the second equation:

\(\displaystyle{e}^{{5}}{b}={16}\)

Take the natural logarithm from each side of the previous equation (also using that the natural logarithm is the inverse of the exponential).

\(\displaystyle{5}{b}={\ln{{16}}}\)

Let us use the power property of logarithms (lna^b=blna) along with \(\displaystyle{16}={2}^{{4}}:\)

\(\displaystyle{5}{b}={4}{\ln{{2}}}\)

Divide each side of the previous equation by 5:

\(\displaystyle{b}=\frac{{{4}{\ln{{2}}}}}{{5}}\)

Replacing a by 1 and b by \(\displaystyle{4}\frac{{\ln{{2}}}}{{5}}\) in the general equation of the exponential model, we then obtain:

\(\displaystyle{y}={e}^{{{4}\frac{{\ln{{2}}}}{{5}}}}{x}\)

General equation of an exponential model: y=aebx

Evaluate the general equation at x=0 and x=5:

\(\displaystyle{y}={a}{e}^{{b}}{\left({0}\right)}={a}{e}^{{0}}={a}{\left({1}\right)}={a}\)

\(\displaystyle{y}={a}{e}^{{b}}{\left({5}\right)}={a}{e}^{{5}}{b}\)

We require y=1 when x=0 and y=16 when x=5 (as the points (0,1) and (5,16) need to lie on the exponential model).

a=1

\(\displaystyle{a}{e}^{{5}}{b}={16}\)

By the first equation, we thus know that a=1. Let us replace \(\displaystyle{a}{e}^{{5}}{b}={16}\) by 1 in the second equation:

\(\displaystyle{e}^{{5}}{b}={16}\)

Take the natural logarithm from each side of the previous equation (also using that the natural logarithm is the inverse of the exponential).

\(\displaystyle{5}{b}={\ln{{16}}}\)

Let us use the power property of logarithms (lna^b=blna) along with \(\displaystyle{16}={2}^{{4}}:\)

\(\displaystyle{5}{b}={4}{\ln{{2}}}\)

Divide each side of the previous equation by 5:

\(\displaystyle{b}=\frac{{{4}{\ln{{2}}}}}{{5}}\)

Replacing a by 1 and b by \(\displaystyle{4}\frac{{\ln{{2}}}}{{5}}\) in the general equation of the exponential model, we then obtain:

\(\displaystyle{y}={e}^{{{4}\frac{{\ln{{2}}}}{{5}}}}{x}\)