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''logarithm''
 
''logarithm''
  
 
The function inverse to the [[Exponential function|exponential function]]. The logarithmic function is denoted by
 
The function inverse to the [[Exponential function|exponential function]]. The logarithmic function is denoted by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l0606001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
=   \mathop{\rm ln}  x ;
 +
$$
  
its value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l0606002.png" />, corresponding to the value of the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l0606003.png" />, is called the natural logarithm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l0606004.png" />. From the definition, relation (1) is equivalent to
+
its value $  y $,  
 +
corresponding to the value of the argument $  x $,  
 +
is called the natural logarithm of $  x $.  
 +
From the definition, relation (1) is equivalent to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l0606005.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
= e  ^ {y} .
 +
$$
  
Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l0606006.png" /> for any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l0606007.png" />, the logarithmic function is defined only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l0606008.png" />. In a more general sense a logarithmic function is a function
+
Since $  e  ^ {y} > 0 $
 +
for any real $  y $,  
 +
the logarithmic function is defined only for $  x > 0 $.  
 +
In a more general sense a logarithmic function is a function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l0606009.png" /></td> </tr></table>
+
$$
 +
=   \mathrm{log} _ {a}  x ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060011.png" />) is an arbitrary base of the logarithm; this function can be expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060012.png" /> by the formula
+
where $  a > 0 $ ($  a \neq 1 $)  
 +
is an arbitrary base of the logarithm; this function can be expressed in terms of $  \mathop{\rm ln}  x $
 +
by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060013.png" /></td> </tr></table>
+
$$
 +
\mathrm{log} _ {a}  x  =
 +
\frac{ \mathop{\rm ln}  x }{ \mathop{\rm ln}  a }
 +
.
 +
$$
  
 
The logarithmic function is one of the main elementary functions; its graph (see Fig.) is called a logarithmic curve.
 
The logarithmic function is one of the main elementary functions; its graph (see Fig.) is called a logarithmic curve.
Line 25: Line 55:
 
The main properties of the logarithmic function follow from the corresponding properties of the exponential function and logarithms; for example, the logarithmic function satisfies the functional equation
 
The main properties of the logarithmic function follow from the corresponding properties of the exponential function and logarithms; for example, the logarithmic function satisfies the functional equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060014.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ln}  x +  \mathop{\rm ln}  y  =   \mathop{\rm ln}  x y .
 +
$$
 +
 
 +
The logarithmic function  $  y = \mathop{\rm ln}  x $
 +
is a strictly-increasing function, and  $  \lim\limits _ {x \downarrow 0 }    \mathop{\rm ln}  x = - \infty $,
 +
$  \lim\limits _ {x \rightarrow \infty }    \mathop{\rm ln}  x = + \infty $.
 +
At every point  $  x > 0 $
 +
the logarithmic function has derivatives of all orders and in a sufficiently small neighbourhood it can be expanded in a power series, that is, it is an [[Analytic function|analytic function]]. For  $  - 1 < x \leq  1 $
 +
the following expansion of the (natural) logarithmic function is valid:
 +
 
 +
$$
 +
\mathop{\rm ln}  ( 1 + x )  =  x -
 +
 
 +
\frac{x  ^ {2}}{2}
 +
+
  
The logarithmic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060015.png" /> is a strictly-increasing function, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060017.png" />. At every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060018.png" /> the logarithmic function has derivatives of all orders and in a sufficiently small neighbourhood it can be expanded in a power series, that is, it is an [[Analytic function|analytic function]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060019.png" /> the following expansion of the (natural) logarithmic function is valid:
+
\frac{x  ^ {3}}{3}
 +
-
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060020.png" /></td> </tr></table>
+
\frac{x  ^ {4}}{4}
 +
+ \dots .
 +
$$
  
 
The derivative of the logarithmic function is
 
The derivative of the logarithmic function is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060021.png" /></td> </tr></table>
+
$$
 +
(  \mathop{\rm ln}  x )  ^  \prime  =
 +
\frac{1}{x}
 +
,\ \
 +
(  \mathrm{log} _ {a}  x )  ^  \prime  = \
 +
 
 +
\frac{ \mathrm{log} _ {a}  e }{x}
 +
  = \
 +
 
 +
\frac{1}{x  \mathop{\rm ln}  a }
 +
.
 +
$$
  
 
Many integrals can be expressed in terms of the logarithmic function; for example:
 
Many integrals can be expressed in terms of the logarithmic function; for example:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060022.png" /></td> </tr></table>
+
$$
 +
\int\limits
 +
\frac{dx}{x}
 +
  =   \mathop{\rm ln}  | x | + C ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060023.png" /></td> </tr></table>
+
$$
 +
\int\limits
 +
\frac{dx}{\sqrt {x  ^ {2} + a } }
 +
  =   \mathop{\rm ln} ( x + \sqrt {x  ^ {2} + a } ) + C .
 +
$$
  
 
The dependence between variable quantities expressed by the logarithmic function was first considered by J. Napier in 1614.
 
The dependence between variable quantities expressed by the logarithmic function was first considered by J. Napier in 1614.
  
The logarithmic function on the complex plane is an infinitely-valued function, defined for all values of the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060024.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060025.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060026.png" /> if no confusion arises). The single-valued branch of this function defined by
+
The logarithmic function on the complex plane is an infinitely-valued function, defined for all values of the argument $  z \neq 0 $,  
 +
and is denoted by $  \mathop{\rm Ln}  z $ (
 +
or $  \mathop{\rm ln}  z $
 +
if no confusion arises). The single-valued branch of this function defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060027.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ln}  z  =   \mathop{\rm ln}  | z | + i  \mathop{\rm arg}  z ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060028.png" /> is the principal value of the argument of the complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060030.png" />, is called the principal value of the logarithmic function. One has
+
where $  \mathop{\rm arg}  z $
 +
is the principal value of the argument of the complex number $  z $,  
 +
$  \pi < \mathop{\rm arg}  z \leq  \pi $,  
 +
is called the principal value of the logarithmic function. One has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060031.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ln}  z  =   \mathop{\rm ln}  z + 2 k \pi i ,\ \
 +
k = 0 , \pm  1 ,\dots .
 +
$$
  
All values of the logarithmic function for negative real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060032.png" /> are purely imaginary complex numbers. The first satisfactory theory of the logarithmic function for complex arguments was given by L. Euler in 1749; he started from the definition
+
All values of the logarithmic function for negative real $  z $
 
+
are purely imaginary complex numbers. The first satisfactory theory of the logarithmic function for complex arguments was given by L. Euler in 1749; he started from the definition
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060033.png" /></td> </tr></table>
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>
 
  
 +
$$
 +
\mathop{\rm Ln}  z  =  \lim\limits _ {n \rightarrow \infty }  n
 +
( z  ^ {1/n} - 1 ) .
 +
$$
  
  
 
====Comments====
 
====Comments====
The principal value of the logarithm maps the punctured complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060034.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060035.png" /> onto the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060036.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060037.png" />-plane. To fill the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060038.png" />-plane one has to map infinitely many copies of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060039.png" />-plane, where for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060040.png" />-th copy one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060042.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060043.png" /> is a [[Branch point|branch point]]. The copies make up the so-called [[Riemann surface|Riemann surface]] of the logarithmic function. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060044.png" /> is a one-to-one mapping of this surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060045.png" /> onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060046.png" />-plane. The derivative of the principal value is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060047.png" /> (as in the real case) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060048.png" />.
+
The principal value of the logarithm maps the punctured complex $  z $-plane $  ( z \neq 0) $
 +
onto the strip $  - \pi < \mathop{\rm Ln}  z \leq  \pi $
 +
in the complex $  w $-plane. To fill the $  w $-plane one has to map infinitely many copies of the $  z $-plane, where for the $  n $-th copy one has $  - \pi + 2 n \pi < \mathop{\rm arg}  z \leq  \pi + 2 n \pi $,
 +
$  n = 0 , \pm  1 ,\dots $.  
 +
In this case 0 $
 +
is a [[Branch point|branch point]]. The copies make up the so-called [[Riemann surface|Riemann surface]] of the logarithmic function. Clearly, $  \mathop{\rm ln}  z $
 +
is a one-to-one mapping of this surface $  ( z \neq 0 ) $
 +
onto the $  w $-plane. The derivative of the principal value is $  1 / z $ (as in the real case) for $  - \pi <  \mathop{\rm arg}  z < \pi $.
  
Instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060050.png" />, many Western writers of post-calculus mathematics use <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060600/l06060052.png" /> (see also (the editorial comments to) [[Logarithm of a number|Logarithm of a number]]).
+
Instead of $  \mathop{\rm ln} $
 +
and $  \mathop{\rm Ln} $,  
 +
many Western writers of post-calculus mathematics use $  \mathop{\rm log} $
 +
and $  \mathop{\rm Log} $ (see also (the editorial comments to) [[Logarithm of a number|Logarithm of a number]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,  "Functions of one complex variable" , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Marsden,  "Basic complex analysis" , Freeman  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of functions" , Oxford Univ. Press  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Saks,  A. Zygmund,  "Analytic functions" , PWN  (1952)  (Translated from Polish)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,  "Functions of one complex variable" , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Marsden,  "Basic complex analysis" , Freeman  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of functions" , Oxford Univ. Press  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Saks,  A. Zygmund,  "Analytic functions" , PWN  (1952)  (Translated from Polish)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR></table>

Latest revision as of 12:10, 1 May 2023


logarithm

The function inverse to the exponential function. The logarithmic function is denoted by

$$ \tag{1 } y = \mathop{\rm ln} x ; $$

its value $ y $, corresponding to the value of the argument $ x $, is called the natural logarithm of $ x $. From the definition, relation (1) is equivalent to

$$ \tag{2 } x = e ^ {y} . $$

Since $ e ^ {y} > 0 $ for any real $ y $, the logarithmic function is defined only for $ x > 0 $. In a more general sense a logarithmic function is a function

$$ y = \mathrm{log} _ {a} x , $$

where $ a > 0 $ ($ a \neq 1 $) is an arbitrary base of the logarithm; this function can be expressed in terms of $ \mathop{\rm ln} x $ by the formula

$$ \mathrm{log} _ {a} x = \frac{ \mathop{\rm ln} x }{ \mathop{\rm ln} a } . $$

The logarithmic function is one of the main elementary functions; its graph (see Fig.) is called a logarithmic curve.

Figure: l060600a

The main properties of the logarithmic function follow from the corresponding properties of the exponential function and logarithms; for example, the logarithmic function satisfies the functional equation

$$ \mathop{\rm ln} x + \mathop{\rm ln} y = \mathop{\rm ln} x y . $$

The logarithmic function $ y = \mathop{\rm ln} x $ is a strictly-increasing function, and $ \lim\limits _ {x \downarrow 0 } \mathop{\rm ln} x = - \infty $, $ \lim\limits _ {x \rightarrow \infty } \mathop{\rm ln} x = + \infty $. At every point $ x > 0 $ the logarithmic function has derivatives of all orders and in a sufficiently small neighbourhood it can be expanded in a power series, that is, it is an analytic function. For $ - 1 < x \leq 1 $ the following expansion of the (natural) logarithmic function is valid:

$$ \mathop{\rm ln} ( 1 + x ) = x - \frac{x ^ {2}}{2} + \frac{x ^ {3}}{3} - \frac{x ^ {4}}{4} + \dots . $$

The derivative of the logarithmic function is

$$ ( \mathop{\rm ln} x ) ^ \prime = \frac{1}{x} ,\ \ ( \mathrm{log} _ {a} x ) ^ \prime = \ \frac{ \mathrm{log} _ {a} e }{x} = \ \frac{1}{x \mathop{\rm ln} a } . $$

Many integrals can be expressed in terms of the logarithmic function; for example:

$$ \int\limits \frac{dx}{x} = \mathop{\rm ln} | x | + C , $$

$$ \int\limits \frac{dx}{\sqrt {x ^ {2} + a } } = \mathop{\rm ln} ( x + \sqrt {x ^ {2} + a } ) + C . $$

The dependence between variable quantities expressed by the logarithmic function was first considered by J. Napier in 1614.

The logarithmic function on the complex plane is an infinitely-valued function, defined for all values of the argument $ z \neq 0 $, and is denoted by $ \mathop{\rm Ln} z $ ( or $ \mathop{\rm ln} z $ if no confusion arises). The single-valued branch of this function defined by

$$ \mathop{\rm ln} z = \mathop{\rm ln} | z | + i \mathop{\rm arg} z , $$

where $ \mathop{\rm arg} z $ is the principal value of the argument of the complex number $ z $, $ \pi < \mathop{\rm arg} z \leq \pi $, is called the principal value of the logarithmic function. One has

$$ \mathop{\rm Ln} z = \mathop{\rm ln} z + 2 k \pi i ,\ \ k = 0 , \pm 1 ,\dots . $$

All values of the logarithmic function for negative real $ z $ are purely imaginary complex numbers. The first satisfactory theory of the logarithmic function for complex arguments was given by L. Euler in 1749; he started from the definition

$$ \mathop{\rm Ln} z = \lim\limits _ {n \rightarrow \infty } n ( z ^ {1/n} - 1 ) . $$


Comments

The principal value of the logarithm maps the punctured complex $ z $-plane $ ( z \neq 0) $ onto the strip $ - \pi < \mathop{\rm Ln} z \leq \pi $ in the complex $ w $-plane. To fill the $ w $-plane one has to map infinitely many copies of the $ z $-plane, where for the $ n $-th copy one has $ - \pi + 2 n \pi < \mathop{\rm arg} z \leq \pi + 2 n \pi $, $ n = 0 , \pm 1 ,\dots $. In this case $ 0 $ is a branch point. The copies make up the so-called Riemann surface of the logarithmic function. Clearly, $ \mathop{\rm ln} z $ is a one-to-one mapping of this surface $ ( z \neq 0 ) $ onto the $ w $-plane. The derivative of the principal value is $ 1 / z $ (as in the real case) for $ - \pi < \mathop{\rm arg} z < \pi $.

Instead of $ \mathop{\rm ln} $ and $ \mathop{\rm Ln} $, many Western writers of post-calculus mathematics use $ \mathop{\rm log} $ and $ \mathop{\rm Log} $ (see also (the editorial comments to) Logarithm of a number).

References

[1] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian)
[2] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[a1] J.B. Conway, "Functions of one complex variable" , Springer (1973)
[a2] E. Marsden, "Basic complex analysis" , Freeman (1973)
[a3] E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979)
[a4] S. Saks, A. Zygmund, "Analytic functions" , PWN (1952) (Translated from Polish)
[a5] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Logarithmic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmic_function&oldid=16249
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article