

Subject:
Natural Logarithms, What are they?
Category: Science > Math Asked by: jscga 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: tillga on 16 Feb 2004 14:01 PST 
I will try to answer as simple as possible without oversimplication. In many natural processes decay functions play an important role. Let us consider the term "halflife" used in the phenomenon of radioactivity. "The measurement of halflives 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 halflife. The logarithm of the decay rate is plotted against time, and a straight line is fitted to the points. The time interval for this straightline decay curve to fall by a factor of 2 is read from the graph as the halflife. 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 straightline 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. tillga Search strategy: Internal search function of the Encylopedia Britannica and personal knowledge as a natural scientist  
 


Subject:
Re: Natural Logarithms, What are they?
From: majortomga on 16 Feb 2004 11:18 PST 
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. 
Subject:
Re: Natural Logarithms, What are they?
From: supermacmanga on 16 Feb 2004 14:32 PST 
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 xintercept of 1 ? has a yintercept 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)  supermacmanga 
Subject:
Re: Natural Logarithms, What are they?
From: farbeyonddrivenga on 23 Feb 2004 23:59 PST 
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. 
If you feel that you have found inappropriate content, please let us know by emailing us at answerssupport@google.com with the question ID listed above. Thank you. 
Search Google Answers for 
Google Home  Answers FAQ  Terms of Service  Privacy Policy 