The function inverse to the exponential function. The logarithmic function is denoted by
its value , corresponding to the value of the argument , is called the natural logarithm of . From the definition, relation (1) is equivalent to
Since for any real , the logarithmic function is defined only for . In a more general sense a logarithmic function is a function
where () is an arbitrary base of the logarithm; this function can be expressed in terms of by the formula
The logarithmic function is one of the main elementary functions; its graph (see Fig.) is called a logarithmic curve.
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 logarithmic function is a strictly-increasing function, and , . At every point 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 the following expansion of the (natural) logarithmic function is valid:
The derivative of the logarithmic function is
Many integrals can be expressed in terms of the logarithmic function; for example:
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 , and is denoted by (or if no confusion arises). The single-valued branch of this function defined by
where is the principal value of the argument of the complex number , , is called the principal value of the logarithmic function. One has
All values of the logarithmic function for negative real 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
|||S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian)|
|||A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)|
The principal value of the logarithm maps the punctured complex -plane onto the strip in the complex -plane. To fill the -plane one has to map infinitely many copies of the -plane, where for the -th copy one has , . In this case is a branch point. The copies make up the so-called Riemann surface of the logarithmic function. Clearly, is a one-to-one mapping of this surface onto the -plane. The derivative of the principal value is (as in the real case) for .
Instead of and , many Western writers of post-calculus mathematics use and (see also (the editorial comments to) Logarithm of a number).
|[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)|
Logarithmic function. BSE-3 (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Logarithmic_function&oldid=16249