natural logarithm

The natural logarithm of a number is the logarithm in base of Euler’s number $e$. It can be defined as the map $\ln :{ℝ}_{+}\to ℝ$ satisfying

$$\ln (x):={\int }_1^x\frac{1}{t}𝑑t.$$ (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 function defined in this way is the inverse of the exponential function. Indeed with equation (1) we have

$$\frac{d}{dx}\ln (e^x)=e^x\cdot \frac{1}{e^x}=1,$$

so there exists $C\in ℝ$ such that

$$\ln (e^x)=x+C.$$

Since $\ln (e^0)=0$ we have $C=0$. One can also prove that the above integral has the defining properties of a logarithm. For example if $x,y\in {ℝ}_{+}$ we have

$\ln (xy)$ $=$ ${\displaystyle {\int }_1^x}{\displaystyle \frac{1}{t}}\mathrm{𝑑𝑡}+{\displaystyle {\int }_x^{xy}}{\displaystyle \frac{1}{t}}\mathrm{𝑑𝑡}$

$=$ $\ln (x)+{\displaystyle {\int }_x^{xy}}{\displaystyle \frac{1}{t}}𝑑t.$

Now applying the substitution law with $u:=\frac{t}{x}$ we have

$${\int }_x^{xy}\frac{1}{t}\mathrm{𝑑𝑡}={\int }_1^y\frac{1}{u}\mathrm{𝑑𝑢}=\ln (y),$$

so we have

$$\ln (xy)=\ln (x)+\ln (y).$$

The natural logarithm can also be represented as a power series around $1$. For we have

$$\ln (1+x)=\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^{k+1}}{k}{x}^k.$$ (2)

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

$$\ln (2)=\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^{k+1}}{k}.$$

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 number $z$ in the form $z=R{e}^{i\phi }$ with $R,\phi \in ℝ$. We could try to define the complex logarithm $\ln (z)$ to be $\ln (R)+i\phi$. However $\phi$ is unique only up to addition of a multiple of $2\pi$. While at first glance this does not appear to be very problematic, it actually prevents one from setting up a continuous logarithm on the complex plane (without 0, where the logarithm should be infinite). Say for example that we let the imaginary part of our logarithm take values from $-\pi$ to $\pi$. Then

$$\underset{\phi \searrow -\pi }{lim}\ln e^{i\phi }=-i\pi$$

and

$$\underset{\phi \nearrow \pi }{lim}\ln e^{i\phi }=i\pi$$

while $e^{i\phi }\to -1$ for both limits. Therefore the logarithm we defined is not continuous at -1. The same argument 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