What is the natural logarithm?

The standard rule for integration is: integrate kx^n dx = kx^(n+1)/(n+1). However, if we try and integrate 1/x in this manner we get, x^0/0, i.e. 1/0, which is infinity. However, if we look at a graph of 1/x, then between two points there is clearly a well defined area, so it must be possible to integrate this. The natural logarithm is a function that we use to do this, whereby ln(a) is the integral of 1/x between 1 and a. It is a logarithmic function with base 'e', where e takes the value of about 2.718, and e^x is known as the exponential function; i.e. it increases at an ever increasing rate. The exponential function is the inverse of the natural logarithm function.