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Manually Calculating Logs and Exponents

Date: 12/16/2001 at 04:32:55
From: Wedge
Subject: Manual calulation of logs and exps

Respected Sir/Madam,

Can you give me a formula to calculate logarithms and exponents of 
base 10?

Date: 12/16/2001 at 05:40:33
From: Doctor Mitteldorf
Subject: Re: Manual calulation of logs and exps

(writing e^x for "e raised to the power x")

There are infinite series representations for e^x and ln(x):

   e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ...

The exclamation points mean "factorial."  4!, or "4 factorial" is the 
product of integers from 1 to 4.

In practice, this is a good way to calculate e^x if x is small, but 
if x is large, you must add up a lot of terms before you get close to 

There are many tricks you can use to make the practical computation 
easier. Here's one:  e^x=10^(x*log(e)), where log(e)=0.43429... If you 
want to take e^x and x is a large number, first multiply x by log(e) 
and find the nearest integer. This gives you the power of ten. For the 
factor that goes in front of the power of 10, you can find e^x for a 
smaller number.

For example, 

   e^20=10^(20*.43429...) = 10^8.6858... = 10^9 * 10*(-.3141...)

You can calculate 10^(-.3142) as e^(-.3141/.43429) = e^-(.7233...)
This way, you can calculate the series for a relatively small number, 
and when you're done, multiply by a billion.

The series for ln(x) is

   ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...

It can be used the same way: calculate it for small numbers only, and 
use the addition property of logarithms so you don't need to calculate 
for large numbers.  

How would you calculate ln(1,000,000) using this formula on only a 
small number?

- Doctor Mitteldorf, The Math Forum   
Associated Topics:
High School Exponents
High School Logs

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