Exponential Functions - Problem 5

A typical problem with exponential functions is, finding the equation of an exponential function that fits certain situations. Here I have an example of a problem like this; find an exponential function f of x equals a times b to the x such that, f of x plus 3 over f of x equals 10 and f of 0 equals 5.

So the first thing you want to do is set this up. Finding an exponential function means finding the a and b values. So you want to set these up using information they give you about f of x plus 3 and f of x. And so f of x plus 3, over f of x is 8 times b to the x plus 3 over f of x is, a times b to the x. A lot of this is going to cancel. A cancels, b to the x plus 3, is b to the x times b to the 3. Find the product of powers property, and so we can see that b to the x is going to cancel as well. So f of x plus 3 over f of x equals 10, what I just discovered is that bq equals 10. Therefore b is a cube root of 10.

Now I just need to find a. I’m going to use this fact here, f of 0 equals 5. F of 0 is a times b to the 0 being the 0 is 1, so a equals 5. And that’s it.

My function is 5 times the base cube root of 10 to the x, a times b to the x.