alternative definition of the natural logarithm

The natural logarithmMathworldPlanetmathPlanetmathPlanetmath functionMathworldPlanetmath ( logx can be defined by an integralDlmfPlanetmath, as shown in the entry to which this entry is attached. However, it can also be defined as the inverse functionMathworldPlanetmath of the exponential functionDlmfDlmfMathworldPlanetmathPlanetmath expx=ex.

In this entry, we show that this definition of logx yields a function that satisfies the logarithm laws logxy=logx+logy and logxr=rlogx hold for any positive real numbers x and y and any real number r. We also show that logx is differentiableMathworldPlanetmathPlanetmath with respect to x on the interval (1,) with derivativePlanetmathPlanetmath 1x. Note that the logarithm laws imply that log1=0. The mean-value theorem implies that these properties characterize the logarithm function.

The proof of the first logarithm law is straightforward. Let x and y be positive real numbers. Then using the fact that ex and logx are inverse functions, we find that


Since ex is an injective function, the equation elogxy=elogx+logy implies the first logarithm law.

For the second logarithm law, observe that


Since ex and logx are inverse functions and ex is differentiable, so is logx. We can use the chain ruleMathworldPlanetmath to find a formulaMathworldPlanetmathPlanetmath for the derivative:


Hence, ddx[logx]=1x.

Title alternative definition of the natural logarithm
Canonical name AlternativeDefinitionOfTheNaturalLogarithm
Date of creation 2013-03-22 16:11:10
Last modified on 2013-03-22 16:11:10
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 17
Author CWoo (3771)
Entry type Topic
Classification msc 97D40
Related topic DerivativeOfExponentialFunction
Related topic DerivativeOfInverseFunction
Related topic Logarithm