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Logarithmic function

Let typeset structure and typeset structure.

The logarithmic function with base typeset structure denoted by typeset structure is defined by typeset structure if typeset structure.  In other words, the logarithmic function is the inverse function to exponential one

f : R -> R, x |-> a^x

which is monotone and thus possesses an inverse.

Here are graphs of logarithmic function with various bases:

[Graphics:HTMLFiles/LogarithmicFunction_8.gif]

Logarithms were discovered independently by Scotsman John Napier in1614, Englishman Henry Briggs in 1617 and by Swiss clockmaker Joost Burgi in 1620. Their objective was to simplify mathematical calculations   [1] .   Neither of them worked with this notion as we do, for instance they did not used a concept of a logarithmic base. Napier defined logarithms as a ratio of two distances in a geometric form (the name logarithm has Greek origin λογος=ratio, αριθμο=number). The possibility to define logarithms as exponents was discovered by John Wallis in 1685 and by Johann Bernoulli in 1694. The notion of the base of the logarithm was introduced by Euler.

The logarithm with base typeset structure, is the most important of the logarithmic functions. The logarithm with base typeset structure is called the natural logarithm, (also hyperbolic logarithm, due to its role in computing area under the hyperbole, see the note below and   . ), and it is denoted ln.  The logarithm with base 10 is also called the common logarithm.

The natural logarithm can also be defined by the integral

Underoverscript[∫, 1, arg3] 1/t d t(1)

Logarithm as a function of a complex variable

The logarithm can also be defined for complex values of the argument, in that case

Log z = ln | z | + i Arg(z) .

To compute sample values of  typeset structure go to .

For instance, typeset structure, if typeset structureand typeset structure. typeset structure is called the main (principal) branch of typeset structure.

The logarithmic function has a branch cut discontinuity in the complex typeset structure plane running from typeset structure to typeset structure.

Logarithm as a function of complex variable is a many-valued function, and as in (1) it can be defined by

Underoverscript[∫, 1, arg3] 1/t d t,

where the path of integration does not go through the origin of the coordinates.

This is the graph of the real part of the principal branch of the logarithmic function together with its graph for real values of the argument

[Graphics:HTMLFiles/LogarithmicFunction_25.gif]

This is the graph of the imaginary value of the logarithmic function

[Graphics:HTMLFiles/LogarithmicFunction_26.gif]

This is the graph of the absolute value of the logarithmic function

[Graphics:HTMLFiles/LogarithmicFunction_27.gif]

Two branches of the Riemann surface of the of the imaginary value of the logarithmic function

[Graphics:HTMLFiles/LogarithmicFunction_28.gif]

Basic properties

Logarithmic functions satisfy the following basic properties:

Series expansions

In 1668, Nicolas Mercator (1620-1687) published  [2]   the well known Taylor series expansion for the logarithmic function (also independently discovered by Grègory Saint-Vincent): 1

ln(1 + z) = z - 1/2 z^2 + 1/3 z^3 - ...         (| z | <= 1, z != 1)(2)

Substitution typeset structure gives

ln 1/(1 - z) = z + 1/2 z^2 + 1/3 z^3 + ...         (| z | <= 1, z != 1)(3)

or that

ln (1 + z)/(1 - z) = 2 (z + 1/2 z^3 + 1/3 z^5 + ... )        (| z | >= 1, z != ± 1)(4)

Inequalities

ln x <= x - 1        (x > 0)(5)
x < -ln(1 - x) < x/(1 - x)          (x < 1, x != 0 )(6)
x/(1 + x) < ln(1 + x) < x       (x > -1, x != 0 )(7)

There follows from (2) that for typeset structure, we have

ln(1 + x) = x + x^2(-1/2 + x/3 - x^2/4 + ...) = x + ϑx^2

where typeset structure using the alternating series error estimate , i.e.

ln(1 + x) = x + x^2 ϑ             (| x | < 1, | ϑ | < 1/2) .(8)

Notes

1 Mecrator used the name logarithmus naturalis (log naturalis) a few decades before the birth of calculus, where the role of this logarithm is mainly used to substantiate the name.

References

[1]  Cajori, F. (1913, No. 1, Jan.). History of the Exponential and Logarithmic Concepts. Amer. Math. Monthly, 20, 5-14.

[2]  Mercator, N. (1668). Logarithmo-technia. London.

Cite this web-page as:

Štefan Porubský: Logarithmic function.

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