## Natural Logairthms

One number raised to the power of another is called a base. In the expression 3^4 the base is 3. The most common base is 10 – we count and measure things in multiples of 10 because we have 10 fingers on which to count. There is however, one base which stands above all others in physical and mathematical significance. This is the basewhere

is irrational and the decimal expansion ofcontinuous forever with no pattern, although well known methods exist for calculatingto however many decimal places are desired.

arises naturally in maths, when the rate of change of something is proportioal to the quantity present.

Suppose the rate of change of a population

If the rate of change ofis proportional towe can writeThis is a differential equation and can be integrated to givewhereis the initial population andis the number given above. . In particular ifthe rate of change of the population is equal to the population and the population will grow by a face ofin unit time period. Whatever the value ofas long asthe growth ofis exponential meaningincreases by a constant factor in each time period (and if the value ofdecreases by a constant factor in each time period).

Logarithms with base e obey the same log rules as all other logs, but the number e is special enough for any log with base e to have a special name. They are called natural logs – or logarithme naturel from French and ln for short - so that

The number e appears in every branch of maths, from number theory, complex numbers, trigonometry, differential equationsm,,, and is one of the most important constants in maths, alongside the numberin significance.