The Natural Logarithm is the Limit of the Integrals of Powers

Assume that and that .

The integral of is , where is an arbitrary constant. The integral of is , where again is an arbitrary constant and is the natural logarithm of , often written as .

When is close to zero, and are close, so there must be some connection between their integrals!

Choose and so that the two integrals are both zero at . The integrals are then and . For close to zero these functions are very close; in symbols, .

Using the difference quotient for the derivative of the base- exponential function with respect to (not ) and using instead of the more usual gives . This is more usually written with as the variable: , with the special case