Applications of LogarithmsDate: 04/15/98 at 04:03:43 From: Hock Heem Subject: Logarithms May I know how can I calculate the logarithm function on a number without a calculator? What are all the uses of logarithm function? Can you please list them? Date: 04/15/98 at 20:17:10 From: Doctor Kate Subject: Re: Logaritms Hock: Here's a great Dr Math answer from the archives that will tell you all about logarithms: http://mathforum.org/dr.math/problems/ojanen.7.31.96.html Also, there's an answer about logarithms without a calculator here: http://mathforum.org/dr.math/problems/eljohnson.8.16.96.html The method I find most interesting (and probably most understandable) is first to notice that: ln (x+1) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + (x^5)/5 ... (continuing forever) for any x between -1 and 1. Well, how do we notice that? I'm afraid I don't know how to explain it without calculus (if you know calculus, write back and I'll try). But it's easy to convince yourself if you try a few. Say you want to know ln 2: ln (1+1) = 1 - 1/2 + 1/3 - 1/4 + 1/5 ... Well ln 2 = 0.693147... according to your calculator, and: 1 = 1 1 - 1/2 = 0.5 1 - 1/2 + 1/3 = 0.8333... 1 - 1/2 + 1/3 - 1/4 = 0.58333... 1 - 1/2 + 1/3 - 1/4 + 1/5 = 0.78333... 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 = 0.61666... So you can see that as we keep adding terms, we get closer and closer to the real value (though pretty slowly). This concept is a big one in mathematics, especially calculus, and it's called 'convergence'. The series 1 - 1/2 + 1/3 - 1/4 + 1/5 ... is an infinite series which converges to ln 2. When you add more and more terms, you get closer and closer to ln 2. In fact, you can get as close as you want just by adding up enough terms. So if you want to calculate a small ln, you can add a lot of terms with this formula. But it takes a long time, so it's best to have a calculator. As for the uses of logarithms in daily life, there are many of them! For example, you can think of money in the bank. When it earns interest, this is a logarithmic process. So is nuclear decay of radioactive substances. So are the rates of chemical reactions. So is the energy you get from a battery over time. In nature, a very large number of natural phenomena take place according to logarithmic or exponential functions. Logarithms are also extremely important in mathematics. I'm afraid there are so many examples that I can't list them all! You are welcome to write back. Hope I've been some help. -Doctor Kate, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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