Natural LogarithmsDate: 11/01/97 at 15:11:07 From: MWaissblut Subject: Natural Logarithms What is "natural" about natural logarithms? Also, why is 'e' a transcendental number? Date: 11/02/97 at 07:28:20 From: Doctor Jerry Subject: Re: Natural Logarithms Hi, I think that there is no single answer as to why natural logarithms are preferred. Here are several possible answers. 1. Just as radians are preferred in differentiation, to avoid constants, the exponential function e^x is preferable to, say, 10^x. The derivative of e^x is, of course, e^x while the derivative of 10^x is (10^x)*ln(10). 2. The differential equation dQ/dt=k*Q giving the rate of decay of a radioactive material separates as dQ/Q = k*dt and then the solution has the form ln(Q)=k*t+c. This is related to the fact that the area beneath the curve y=1/x, from x=1 to x=t, is ln(t). 3. Related to number 2, if one writes the expression for the principal at time t of an amount invested at r percent, compounded k times per year, and takes the limit as k becomes infinite (to obtain continuous compounding), it happens that one obtains an expression of the form (1+1/m)^m. The question is what is the value of this expression as m becomes infinite. The answer is e. The point is that the number e occurs "naturally" in calculations dealing with growth, just as pi occurs naturally in connection with circles or spheres. As to why e is transcendental? Why is pi transcendental? To give an answer runs the risk of going philosophical, which, in my opinion, rarely leads to useful thoughts. However, there would seem to be no reason that pi should turn out to be 22/7, for example. Pi can be thought of as the result of any of several infinite processes (inscribing polygons in a unit circle, for example) and I think one would be surprised if a rational number resulted. Since the exponential function and trig functions are so strongly related, it would be surprising if e were to be rational. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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