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### Applications of Logarithms

```
Date: 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
```

```
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

http://mathforum.org/dr.math/problems/ojanen.7.31.96.html

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

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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