Natural Logarithm

Concept Problem 1 Problem 2 Problem 3

The natural log is the logarithm to the base of the number e and is the inverse function of an exponential function. Natural logarithms are special types of logarithms and are used in solving time and growth problems. Logarithmic functions and exponential functions are the foundations of logarithms and natural logs.

Okay. I wanted to talk about the number E and why we use it as a base for exponential functions, why is it so special. So I've drawn a picture here. The graph of an arbitrary exponential function F of X equals B to the X, that's this graph in purple. And I've also drawn its tangent line at the .01 Now what I'd like to do is I'd like to explore the relationship between the base B and the slope of the tangent line. In order to do that, I'm going to use a demonstration from Geometer Sketch Pad. Okay. So you can see I've got graphed the function G of X equals 2 to the X here. It's actually B to the X. But I can change the value of B to any value I want. That graph is in red. And the graph, the tangent line, is in blue. Right now the slope is .693 Let me move this tangent line around. Notice as I move the tangent line it's still tangent at the .01, But as I move the tangent line around, the base changes. As I move it to the tangent line is less steep, the base gets smaller. As I move it to the tangent line is more steep, the base gets bigger. And if I moved it so that the tangent line had a negative slope, the bases between 0 and 1. Okay. Let's take a look at some particular values. When B equals 2. Again, the slope is .693, When B is 3, the slope is 1.099. So that makes me wonder where is the slope equal to one? Is it 2.5? No. 2.75? No. It turns out that if I want to get the slope to be exactly 1, I need B to be 2.71828. It's this number E. It's the only base that will make it so that the tangent line has a slope of exactly 1.01. Okay. So let's summarize what we discovered. If B is greater than 1, then the slope of the tangent line is positive. If B is between 0 and 1, then the slope of the tangent is negative. If you want the slope to be exactly 1, you need B to equal E. And E is approximately 2.71828, So that is a little glimpse into what makes the number E special. Now let me give you a definition for the number E. E has a very complicated definition. It's a limit as N approaches infinity of 1 plus 1 over N to the N. Now, to help you understand this definition a little bit better, I'm going to calculate some values for this expression 1 plus 1 over N to the N. So I'll make a little table. Let me start with the value 1. When N equals 1, I get 1 plus 1 over 1, 2, to the 1. So I get 2. And anything past that I'm going to need my calculator. So when I plug in 10, I'm getting 1 plus 1 over 10 to the 10th power, according to my calculator it's approximately 2.5937. If I plug in 100, I get 1 plus 1 over 100 to the hundredeth power. It's approximately 2.7048. I'm going to keep going up by powers of 10. So a thousand, I get -- I'm not going to write this out anymore. 2.71692. How about a million? 2.71828. So you finally get some convergence once you get N out to a million. It takes quite a while for this limit, for this limit to start getting really close, for this value to start getting really close to E. But remember that E is defined as the limit of this expression. So the value, these values are heading towards E as N goes to infinity.