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Base of Common Logarithms

Date: 05/23/2000 at 20:57:42
From: Courtney Shannon
Subject: Logarithms and base ten

Why is the base of common logarithms 10? Is it easier to work with, or 
is there a mathematical explanation?

Date: 05/24/2000 at 12:48:28
From: Doctor Peterson
Subject: Re: Logarithms and base ten

Hi, Courtney.

When logarithms were used to just make calculations easier, it was 
natural to use base 10 logs because they fit well with base 10 
numbers. It's easy to find the log of 10, 100, 1000, and so on; and if 
you have a table of logs of numbers from 1 to 10, you can use nothing 
more than addition to use it for larger or smaller numbers:

     log(5)  = 0.69897
     log(50) = log(10*5) = log(10) + log(5) = 1 + 0.69897
             = 1.69897

So common logarithms make it easier to make and use a table. That's 
really all they're good for - except to determine how many digits a 
number has (in base 10, of course).

Mathematically, common logs are NOT particularly "natural"; we usually 
work with "natural logs" (base e), which arise naturally in calculus, 
or else with whatever log fits the work we are doing. Base 2 logs are 
useful in working with binary numbers, for example.

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

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