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Why logarithms?

Tkačik Š

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2007

Vol. 12

429-434

In 16th and 17th century, the need for speed in complex calculation spurred the invention of a powerful mathematical tool known as LOGARITHM. The reduction of multiplication and division to addition and subtraction (likewise the reduction of a complex mathematical structure to more simple ones) is in the spirit of „prosthaphaeretic rules” of ancient Greeks. We discuss some mathematical ideas related to logarithms and present some historical notes.

On a characterization of the logarithm by a mean value property

Ger R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

1998

Vol. 4

1-7

Any real polynomial f(x) = ax2 + bx + c, x ∈ IR, has the property that f (x)-f (y) x-y for every (x, y) ∈ IR, x ꞊ y. It turns out that that particular form of the Lagrange mean value theorem characterizes polynomials of at most second degree. Much more can be proved: J. Aczél [1] has shown that, with no regularity assumptions, a triple (/, g, h) of functions mapping IR into itself satisfies the equation f(x)-g(y) x-y= h(x + y) for all (x, y) ∈ IR, x ≠ y, if and only if there exist real constants a, 6, c such that f (x) = g(x) = ax2 + b, x + c, x ∈ IR, and h(x) = ax + b, x ∈ IR. Generalizations involving weighted arithmetic means were also considered (see e.g. M. Falkowitz [3] and the references therein) and characterizations of polynomials of higher degrees (in the same spirit) were obtained (see [4] and [5], for instance). In what follows we are going to characterize the logarithm in a similar way. To this end, denote by D the open first quadrant of the real plane IR2 with the diagonal removed, i.e. D := (O, ∞)2 \ {(x, x) e IR2 : x ∈ (0, ∞) }.Applying the classical Lagrange mean value theorem to the logaritmic function we derive the existence of a function D 3 (x, y) -> £(x,y) € intcony {x, y} such that the equality log a:-log y x-y £(z,y)

Linear combinations of right invertible operators in commutative algebras with logarithms

Przeworska-Rolewicz D.

Demonstratio Mathematica

1998

Vol. 31, nr 4

887-898

It is well known that a power of a right invertible operators is again right invertible, as well as a polynomial in a right invertible operator under appropriate assumptions. However , a linear combination of right invertible operators (in particular , their sum and/or difference) in general is not right invertible. It will be shown how to solve equations with linear combinations of right invertible operators in commutative algebras using properties of logarithmic and antilogarithmic mappings. The used method is, in a sense, a kind of the variables separation method. We shall obtain also an analogue of the classical Fourier method for partial differential equations. Note that results concerning the Fourier method are proved under weaker assumptions than those obtained in PR[l] (cf. also PR[2]). The extensive bibliography of the subject can be found in PR[2] and PR[4].