PlanetMath: natural logarithm

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natural logarithm

(Definition)

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

$\displaystyle \ln(x)\colon =\int_1^x\frac{1}{t}dt.$

(1)

Figure 1 shows the graph of $\ln$ .

**Figure:** 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

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

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

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

Since $\ln(e^0)=0$ we have

. 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

$\displaystyle \ln(xy)$	$\displaystyle =$	$\displaystyle \int_1^x\frac{1}{t}\mathit{dt}+\int_x^{xy}\frac{1}{t}\mathit{dt}$
	$\displaystyle =$	$\displaystyle \ln(x)+\int_x^{xy}\frac{1}{t}dt.$

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

$\displaystyle \int_x^{xy}\frac{1}{t}\mathit{dt}=\int_1^y\frac{1}{u}\mathit{du}=\ln(y),$

so we have

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

The natural logarithm can also be represented as a power series around

. For $-1<x\le 1$ we have

$\displaystyle \ln(1+x)=\sum_{k=0}^\infty \frac{(-1)^{k+1}}{k}x^k.$

(2)

This series is divergent at

but for

we have convergence due to Leibniz's theorem and obtain

$\displaystyle \ln(2)=\sum_{k=0}^\infty \frac{(-1)^{k+1}}{k}.$

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

in the form $z=Re^{i\varphi}$ with $R,\varphi\in\mathbb{R}$ . We could try to define the complex logarithm $\ln(z)$ to be $\ln(R)+i\varphi$ . However $\varphi$ 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

$\displaystyle \lim_{\varphi\searrow-\pi}\ln e^{i\varphi}=-i\pi$

and

$\displaystyle \lim_{\varphi\nearrow\pi}\ln e^{i\varphi}=i\pi$

while $e^{i\varphi}\to-1$ 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.

"natural logarithm" is owned by mathwizard. [ full author list (2) | owner history (1) ]

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See Also: matrix logarithm, complex logarithm

Attachments:: alternative definition of the natural logarithm (Topic) by CWoo

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Cross-references: plane, negative, argument, continuous at, limits, imaginary part, infinite, complex plane, continuous, multiple, addition, complex number, Euler relation, complex analysis, negative numbers, real, Leibniz's theorem, divergent, series, power series, defining properties, integral, equation, exponential function, inverse, function, calculators, graph, map, Euler's number, base, logarithm
There are 9 references to this entry.

This is version 9 of natural logarithm, born on 2002-02-25, modified 2006-10-02.
Object id is 2666, canonical name is NaturalLogarithm2.
Accessed 19790 times total.

Classification:

AMS MSC:

33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions)

Pending Errata and Addenda

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