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

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