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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