## 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 base where  is irrational and the decimal expansion of continuous forever with no pattern, although well known methods exist for calculating to 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 of is proportional to we can write This is a differential equation and can be integrated to give where is the initial population and is the number given above. . In particular if the rate of change of the population is equal to the population and the population will grow by a face of in unit time period. Whatever the value of as long as the growth of is exponential meaning increases by a constant factor in each time period (and if the value of decreases 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 number in significance. 