chris' question at Yahoo! Answers regarding an exponential function

MarkFL

Staff member
Here is the question:

f(0)=3, f(3)=4 ; how can I solve this? I keep on getting the wrong answers..
I have posted a link there to this topic so the OP can see my work.

MarkFL

Staff member
Hello chris,

We are given:

$$\displaystyle f(x)=ae^{bx}$$

where:

$$\displaystyle f(0)=3,=,f(3)=4$$

Using the first point, we find:

$$\displaystyle f(0)=ae^{b\cdot0}=a=3$$

Now, using this value for $a$, and the second point, we find:

$$\displaystyle f(3)=3e^{b\cdot3}=4\implies e^{3b}=\frac{4}{3}\implies b=\frac{1}{3}\ln\left(\frac{4}{3} \right)$$

And so our function is:

$$\displaystyle f(x)=3e^{\frac{1}{3}\ln\left(\frac{4}{3} \right)x}$$

Although this is the form required, we could rewrite it in a simpler form:

$$\displaystyle f(x)=3e^{\ln\left(\left(\frac{4}{3} \right)^{\frac{x}{3}} \right)}=3\left(\frac{4}{3} \right)^{\frac{x}{3}}$$