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The Number e and the Natural Logarithm - Problem 3
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I want to talk about the family of functions f of x equals e to the rx. Let me do a little bit of algebra here.
I can separate the e to the rx using the power to a power property. And when I do, I realize that this is just an ordinary exponential function where e to the r is my b, my base. And so if r is bigger than 0, then the base is going to be bigger than 1. And if r is less than 0, then b is going to be between 0 and 1. What this means is, you can describe any exponential function with base e and that's often what you’ll see in more advanced courses. Exponential functions will look like this. They’ll write e to some coefficient times x.
And so what I want to do right now is I want to practice changing from some other base to base e. In order to do this, we'll call the property e to the lnx equals x. This is one of the inverse properties of natural log and e to the x.
So here is my first example; change to base e f of x equals 1.06 to the x. So the first thing I want to do is I want to rewrite this as e to the ln of 1.06. Because e to the lnx equals x, e to the ln 1.06 equals 1.06, I’m just using it in reverse.
So this is the same as e to the ln of 1.06 times x using the power to a power rule. And that’s approximately if you wanted the decimal coefficient. Ln of 1.06 is about .0583, so e to the .0583x. So this is in the form e to the rx. So I’ve changed from some arbitrate base 1.06 to e to the rx and you can do that with any base. Let’s do it with this one, .87 to the x. Same trick, you use this inverse property of exponential functions. .87 is going to equals e to the ln of .87 and again using the power to a power property e to the ln of .87 times x.
Once again this is going to equal and I’ll calculate ln of .87, it's approximately -.139x. So as I said before, when your b value is bigger than 1, you are going to get a positive r value. When your b value is less than 1, between 0 and 1 you are going to get a negative value. This is what exponential growth looks like, this is what exponential decay looks like. This function will decrease, this one will decrease.
It's very easy to see that when you have it written as base e. A positive coefficient means an increase in the exponential function. A negative coefficient means a decrease in the exponential function.
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