Logarithms

 

INTRODUCTION

 

Logarithms were historically a simplified way to manually multiply two large numbers and especially to approximate numbers raised to non-integral powers.  But with the advent of electronic hand calculators, that rationale has largely gone away.  Nevertheless they remain an important tool of mathematics.

 

If we have a number raised to some power, that is we have a base with an exponent, we can write the equation

 

 

where the “base” is a real number greater than zero and not equal to one and the “exponent” is any real number.  Then we can define a quantity called the “logarithm” which is nothing more than another way to write the exponent, as

 

 

This is the definition of a logarithm which is simply the exponent which when applied to some base returns a value.  We can then rearrange terms to write an important corollary as

 

 

BASIC FORMULAS

 

For manual calculations, one would first compute the logarithm of “X” and “Y”, do the multiplications or divisions, and then compute the anti-logarithm of the result, as follows:

 

 

 

 

USEFUL EXTENSIONS

 

An interesting relation is

 

 

And this is because we can apply both sides of the above equation as an exponent on the base “a” as follows

 

 

Or if we want to change the base of a logarithm, we could use

 

 

Or we could rewrite this equation to compute the logarithm to an arbitrary base as

 

 

For the logarithm of a sum, we can write

 

 

SPECIFIC VALUES