Suppose we want to find the equation of the exponential function with initital value
\(2\) and goes through the point
\((3,10)\text{.}\) We are still working with an exponential function, so we use the same template.
\begin{equation*}
f(x)=a\cdot b^x
\end{equation*}
Since we know the initial value is
\(2\text{,}\) we can fill that in right away.
\begin{equation*}
f(x)=2\cdot b^x
\end{equation*}
Now, in the previous example, we were given a rate to be able to find
\(b\text{.}\) In this case, we donβt know the rate, but we do have a point on the graph. So, we will use a strategy we have seen lots of times before: plug in the values for
\(x\) and
\(y\) and solve for the thing we donβt know.
\begin{align*}
10\amp=2\cdot b^3\\
5\amp= b^3\\
\sqrt[3]{5}\amp=b
\end{align*}
So, we have
\(b=\sqrt[3]{5}\text{,}\) which we can also write as
\(5^{1/3}\text{.}\) Filling this in, we have our final answer.
\begin{equation*}
f(x)=2\left(\sqrt[3]{5}\right)^x
\end{equation*}