Exponential and Logarithmic Functions
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
One of the problems you are going to see a lot of when you are studying exponential functions is, find the exponential function that matches the situation. Well here, the situation is a graph. We want to find an exponential function f of x equals 8 times b to the x that matches this graph.
So the first thing you want to do is you have to know the formula of an exponential function. You have to know this form. And if you know that the graph passes through these points, you can use that to figure out what the coefficients a and b are.
So for example, using the point (1,54) f of x is going to be 54, and x is going to be 1. So 54 equals 8 times b to the 1. That’s one equation. And using a second point, f of x is going to be 24 and x is going to be 3. So you have two equations with 2 unknowns. Easy to solve, all I need to do is take one equation, and divide both sides by the other equation. This is kind of like the method of elimination.
And you see that a lot of cancellation that’s going to happen here. There a’s will cancel, some of the b’s will cancel, and I can also reduce this fraction; 24 over 54. 24 is 2 times 12, this is 2 times 27. So I have 12 over 27, but that reduces even any further. Both of these have a factor of 3. 3 times 4 over 3 times 9, cancel the 3’s. So b² is 4 over 9. That means b is plus or minus 2/3. Now we know that the base of an exponential function has to be positive. So b has to be 2/3. All I have to do is find a.
Let’s use this equation because I'm going to have to raise b to the first power. So it will be a little easier to find a. 54 equals a times b. Equals a times 2/3. Now I just multiply both sides by 3/2. I get 3 times 27, 81 equals a, a equals 81. And so my function is f of x equals 81 times 2/3 to the x.
That’s it. All you have to do when you are finding the equation of an exponential function is, use the points that you are given, (1,54) and (3,24), to set up equations that involve the variables that are the coefficients of your exponential function and solve for them.