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 Subject: Natural Logarithms, What are they? Category: Science > Math Asked by: jsc-ga List Price: \$5.00 Posted: 16 Feb 2004 09:45 PST Expires: 17 Mar 2004 09:45 PST Question ID: 307324
 ```I am being asked what a natural logarithm is? I have search every resource i have, but have not found a simple explination. I know the important stuff - inverse on e^x, base of e, etc. My calculus students want to know what it is and why it is? HELP!! In simple terms.```
 Subject: Re: Natural Logarithms, What are they? Answered By: till-ga on 16 Feb 2004 14:01 PST
 ```I will try to answer as simple as possible without over-simplication. In many natural processes decay functions play an important role. Let us consider the term "half-life" used in the phenomenon of radioactivity. "The measurement of half-lives of radioactivity in the range of seconds to a few years commonly involves measuring the intensity of radiation at successive times over a time range comparable to the half-life. The logarithm of the decay rate is plotted against time, and a straight line is fitted to the points. The time interval for this straight-line decay curve to fall by a factor of 2 is read from the graph as the half-life. If there is more than one activity present in the sample, the decay curve will not be a straight line over its entire length, but it should be resolvable graphically (or by more sophisticated statistical analysis) into sums and differences of straight-line exponential terms." from: ( The Encyclopedai Britannica CD Rom Deluxe Edition 2001 ) The formulas to describe this are (1) -dN/dt =lambda N (with lamda = decay constant, N = amount of particles and t=time which can be solved like this: (2) N(t)= N(0)e^-(lambda t) Thatīs why the logarithms are called natural logarithms. They appear in many natural processes. There are many more examples where processes that can be described in similar formula using exponential functions. Examples are transportation of heat through a wall, growth of plants or bacteria and many more. I hope this helps to answer your question. If you need more information please post a clarification request before rating my answer. till-ga Search strategy: Internal search function of the Encylopedia Britannica and personal knowledge as a natural scientist``` Request for Answer Clarification by jsc-ga on 16 Feb 2004 16:34 PST ```I think that is a excellent example of how logarithms (actually, exponential equations are used in life), but I am pretty sure that natural logarithms got there name for other reasons than being used in natural processes. It still does not answer what a natural logarithm (ln) is and why it exists. Have you found anything else? Thanks.``` Clarification of Answer by till-ga on 16 Feb 2004 23:48 PST ```All sources available state that there is indeed one main reason for the name "natural". As already mentioned by majortom-ga in his comment the source ( http://en.wikipedia.org/wiki/Natural_logarithm ) mentions that there can be another reason besides the one mentioned: "What's so "natural" about them? Initially, it seems that in a world using base 10 for nearly all calculations, this base would be more "natural" than base e. The reason we call ln(x) "natural" is twofold: first, the natural logarithm can be defined quite easily using a simple integral or Taylor series as will be explained below; this is not true of other logarithms. Second, expressions in which the unknown variable appears as the exponent of e occur much more often than exponents of 10 (because of the "natural" properties of the exponential function which allow to describe growth and decay behaviors), and so the natural logarithm is more useful in practice." The Taylor series leading to the number "e" is given at: ( http://en.wikipedia.org/math/63c083c45a4b64f2f123047dda84f66a.png ) Another source suporting the main reason for the name: "Natural logarithms and exponentials appear in a tremendous number of mathematical models for real world phenomena; you will see them frequently in engineering" ( http://www.ces.clemson.edu/ge/staff/park/Class/ENGR120/Handouts/Logarithms.html ) Sorry, but even with another extended search I canīt find any other reason for the name "natural". Besides I doubt if it will be possible to answer the question "and why it exists" you posted in your clarification request". Itīs a more philosophical question to find out WHY natural phenomena exist. Again I hope that now you are satisfied with my answer. till-ga Search strategy ( ://www.google.de/search?q=%22natural+logarithm%22+name+reason&btnG=Google+Suche&hl=de&ie=UTF-8&oe=UTF-8 )```
 ```The Wikipedia has a fine entry on this, including a paragraph that directly addresses the question, but I can't seem to phrase it clearly enough myself, so I'm going to admit my own understanding is limited and give you the benefit of my research: Natural logarithm - Wikipedia http://en.wikipedia.org/wiki/Natural_logarithm I am sure this will be helpful.```
 ```Here is a precise mathematical definition. Natural logarithms - a) in the curve y = a^x, a = e when dy/dx = a^x. In other words, the curve of y = e^x has a slope of e^x for all values of x. The inverse of y = e^x is ln y = x. So, x = ln y is y = e^x relfected in the line y = x. b) the integral of 1/x. So, if dy = dx/x, y = ln |x| + C. c) integral (no pun intended) to all natural systems, rates of growth, population cycles, electricity, capacitance, magnetism. d) y = ln x has the following properties: ? is defined for all values x > 0 ? has a range of y is an element of all real numbers ? is concave down for x > 0 ? is increasing for x > 0 ? has dy/dx = 1/x ? has an x-intercept of 1 ? has a y-intercept of negative infinity ? is converging for x in the interval (0, 1] ? can be written as y = log(e) x (i.e. logarithm with base e) - supermacman-ga```
 ```This may or may not be touched on already, I didn't read the posts in their entirety. Here's as simple and concise a reason as possible: Derivatives of exponential functions are proportional to the function itself, or in symbols: If y = b^x, then dy/dx = kb^x = ky , with k = lim [{b^h - 1}/h] as h extends to infinity (this is based on applying the explicit definition of the derivitive to the find the derivative of the function b^x) This being the case, the question arose as to what value of B would make the above equation especially convenient. Choosing a B to make the proportionality constant k equal 1 would the simplest (or most "natural") choice. So what value of b makes k = 1? None other than e. So if the number e is chosen as the base, then the exponential function is equal to its own derivative. This is the only function that has this property.```