 The natural logarithmic function log x gives the log to base e of x, where e is approximately two point mumble. Earlier we were told that the derivative of e to the x was e to the x because the derivative of log was one over x. Or maybe you were told that the derivative of log was one over x because the derivative of e to the x was e to the x. But finding either of these derivatives from the definition of the derivative is impossible. And why is two point seven mumble a natural base? We'll answer these questions using a technique drawn from the history of mathematics. Around 1658 Pierre de Fermat described a method of finding the areas under all curves of the form y equals x to the n, where n was any positive or negative rational number, except for n equals negative one. We can still apply Fermat's method to find the area under the curve. Let L of t be the area bounded by the graph of y equals one over x and the x axis over some interval. We'll choose some n and let r be the nth root of a, then partition the interval using the geometric sequence one r, r squared, r cubed, and so on up to r to the n, which will be equal to a, and then form our upper or lower or left or right rectangle. We'll start with the lower rectangles. The first rectangle has with r minus one and height one over r, so its area is the second rectangle has with r squared minus r, which is r times r minus one and height one over r squared, giving area. The third rectangle has with r cubed minus r squared, which will factor is r squared r minus one, and height one over r cubed, so its area will be. And there's n of these rectangles, so the total area of all the rectangles will be. And since r is the nth root of a, we can rewrite this without having to rely on r. Assuming the limit exists and equals the area, this gives us the following result. For a greater than or equal to one, l of a is strictly greater than, for all whole numbers n, with l of a the limit of the expression as n goes to infinity. All we need to do is find the limit as n goes to infinity. Unfortunately, none of our usual methods allow us to find the limit. Now, not knowing the answer would probably stop a normal person, but mathematicians are not normal people. And so a useful idea to keep in mind for math and life is continue, even if you don't know the answer. In this case we're actually interested in the properties of l of t, so for now we don't need to know what the limit actually is. Now consider any b greater than zero, and consider the area over the interval from b to b times a. Again, we'll let r be the nth root of a, and partition our interval at the points b, b r, b r squared, and so on up to b r to the n, which again, because r is the nth root of a, that's just b a, and we'll form our rectangles. So the first rectangle has height 1 over b r, and with b r minus b, that's b times r minus 1, so its area will be, the second rectangle has height 1 over b r squared, and with b r squared minus b r, that's b r times r minus 1, so its area will be, the third rectangle has height 1 over b r cubed, and with b r cubed minus b r squared, that's b r squared times r minus 1, so its area will be, and so the total area of all the rectangles will be, and again we can rewrite that using r as the nth root of a, but this is just l of a again, giving us the result for b greater than zero, the area under y equals 1 over x over the interval from b to b a is just l of a. Now consider l of a b, this is the area under y equals 1 over x over the interval from 1 to a b, but this area is the same as the area over the interval from 1 to b, well that's just l of b, plus the area over the interval from b to b a, but this is the same as l of a, and so l of a b is l of a plus l of b, and so l of t is a function that has the property that l of the product a b is the sum of l of a plus l of b for all a and b greater than 1, but this is the property that logarithmic functions have, and remember things that do the same thing are the same thing, since this area function has the same properties that the logarithmic function has, l of t is a logarithmic function with some base. Now note all the preceding holds for the area of any curve of the form y equals k over x, where k is greater than zero, but y equals 1 over x is the simplest such curve, so we'll identify the area function for this curve as a natural logarithmic function, where l of t is log t and our integral from 1 to a of 1 over t dt is log of a. Now we designate the base of the natural log using e, but what's the value of e? Since e is the base of our log function, we note that log of e is equal to 1, and so that gives us a way to find e as follows. From our area function being the limit, we want the area to be 1 for some value e. Now for all n, the total area of all the rectangles will always be less than the actual area, so we have our inequality, and we could do a little bit of algebra, and this gives us an upper bound for e. As for a lower bound, suppose we use the upper rectangles to find our area. Then the first rectangle would have height 1 and width r minus 1 for area. The second rectangle would have height 1 over r and width r squared minus r for area. The third rectangle would height 1 over r squared and width r cubed minus r squared for area, and so on. So the upper sum would be, and this would always be greater than 1. So again, doing a little bit of algebra, we can find a lower bound for e, and so e is between two expressions, and we'll apply the squeeze theorem backwards. Ordinarily we'd use the squeeze theorem to find a limit, but we know the limits exist and they're both equal to e, and so we can define e as the limit as n goes to infinity of, and while we could do this, because e is either limit, we usually define e to be this limit.