# 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 $\mathrm{ln}:{\mathbb{R}}_{+}\to \mathbb{R}$ satisfying

$$\mathrm{ln}(x):={\int}_{1}^{x}\frac{1}{t}\mathit{d}t.$$ | (1) |

Figure 1 shows the graph of $\mathrm{ln}$.

Instead of $\mathrm{ln}$ many mathematicians write $\mathrm{log}$, physicists (and calculators) however consider $\mathrm{log}$ 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

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

so there exists $C\in \mathbb{R}$ such that

$$\mathrm{ln}({e}^{x})=x+C.$$ |

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

$\mathrm{ln}(xy)$ | $=$ | $\int}_{1}^{x}}{\displaystyle \frac{1}{t}}\mathrm{\mathit{d}\mathit{t}}+{\displaystyle {\int}_{x}^{xy}}{\displaystyle \frac{1}{t}}\mathrm{\mathit{d}\mathit{t$ | ||

$=$ | $\mathrm{ln}(x)+{\displaystyle {\int}_{x}^{xy}}{\displaystyle \frac{1}{t}}\mathit{d}t.$ |

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

$${\int}_{x}^{xy}\frac{1}{t}\mathrm{\mathit{d}\mathit{t}}={\int}_{1}^{y}\frac{1}{u}\mathrm{\mathit{d}\mathit{u}}=\mathrm{ln}(y),$$ |

so we have

$$\mathrm{ln}(xy)=\mathrm{ln}(x)+\mathrm{ln}(y).$$ |

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

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

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

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

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 $R,\phi \in \mathbb{R}$.
We could try to define the complex logarithm $\mathrm{ln}(z)$ to be $\mathrm{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

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

and

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

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 |