Logarithmic function

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)