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

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

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)

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

[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)