Logarithms and Base EDate: 11/20/97 at 00:31:33 From: Steven Subject: Logarithms and base e I cannot figure out why the base of a natural logarithm is "e" - I have asked my professor, but he was leaving for the day and said that "it is a long and complicated process to prove why it works." My question was how did "e" recive a value of 2.17... ? Can you help me? Date: 11/24/97 at 16:05:43 From: Doctor Bruce Subject: Re: Logarithms and base e Hello Steven, No doubt you are familiar with common logarithms, which are to the base 10. These seem natural enough to most poeple, since we are familiar with arithmetic in base 10. It sounds as if you want to know where the very strange-looking number 2.7182818284 ... comes from, and why we mathematicians think it is so much more natural than good old 10. If we plot the graph of the common logarithm function y = log_10(x) and take some very careful measurements, we find that the slope of the tangent line at the point (1,0) is approximately equal to .434. We can plot the logarithm function with other bases, too. For example, if we plot y = log_2(x), we find the slope of the tangent at (1,0) is approximately equal to 1.44. If we plot y = log_3(x), we find the slope of the tangent at (1,0) is approximately equal to .91. So, we ask, is there a value between 2 and 3 which could serve as the base for logarithms, such that the tangent at (1,0) has slope exactly equal to 1? The answer is yes, that value is 2.718 ..., which we typically call "e". If you know some calculus, in particular, how to take derivatives, then you know that the derivative of log_e(x) is 1/x. But for any other base of logarithms, the derivative will be 1/x multiplied by a number other than 1. The derivative of log_2(x) is (1.44...)/x, for example. That's why using e is so natural. The value of e is easily computed to as many decimals as you like by adding up a few terms of the series e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ... -Doctor Bruce, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 11/25/97 at 13:42:24 From: Anonymous Subject: Re: Logarithms and base e Hey thanks, now I know where e got its value. |
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