natural logarithm

The natural logarithmMathworldPlanetmathPlanetmathPlanetmath of a number is the logarithm in base of Euler’s number e. It can be defined as the map ln:+ satisfying

ln(x):=1x1t𝑑t. (1)

Figure 1 shows the graph of ln.

Figure 1: The graph of ln(x).

Instead of ln many mathematicians write log, physicists (and calculators) however consider log as the symbol for the logarithm in base 10. One can show that the functionMathworldPlanetmath defined in this way is the inversePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of the exponential functionDlmfDlmfMathworldPlanetmathPlanetmath. Indeed with equation (1) we have


so there exists C such that


Since ln(e0)=0 we have C=0. One can also prove that the above integralDlmfPlanetmath has the defining properties of a logarithm. For example if x,y+ we have

ln(xy) = 1x1t𝑑𝑡+xxy1t𝑑𝑡
= ln(x)+xxy1t𝑑t.

Now applying the substitution law with u:=tx we have


so we have


The natural logarithm can also be represented as a power seriesMathworldPlanetmath around 1. For -1<x1 we have

ln(1+x)=k=1(-1)k+1kxk. (2)

This series is divergent at x=-1 but for x=1 we have convergence due to Leibniz’s theoremMathworldPlanetmath and obtain


In real analysis there is no reasonable way to extend the logarithm to negative numbers. In complex analysis the situation is a bit more complicated. Basically one can use the Euler relation to write a non-zero complex numberMathworldPlanetmathPlanetmath z in the form z=Reiφ with R,φ. We could try to define the complex logarithm ln(z) to be ln(R)+iφ. However φ is unique only up to additionPlanetmathPlanetmath of a multiple of 2π. While at first glance this does not appear to be very problematic, it actually prevents one from setting up a continuousMathworldPlanetmathPlanetmath logarithm on the complex planeMathworldPlanetmath (without 0, where the logarithm should be infiniteMathworldPlanetmathPlanetmath). Say for example that we let the imaginary partDlmfMathworld of our logarithm take values from -π to π. Then




while eiφ-1 for both limits. Therefore the logarithm we defined is not continuous at -1. The same argumentMathworldPlanetmath allows one to show that it is not continuous on the negative real numbers. In fact you can only define a continuous complex logarithm on a sliced plane, i.e. the complex plane with a half-line starting at 0 removed.

Title natural logarithm
Canonical name NaturalLogarithm
Date of creation 2013-03-22 12:28:28
Last modified on 2013-03-22 12:28:28
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 13
Author mathwizard (128)
Entry type Definition
Classification msc 33B10
Related topic MatrixLogarithm
Related topic ComplexLogarithm