Logaritmilt

Logaritmilt is a term used in mathematical thought experiments to denote a family of monotone transforms on the positive real numbers parameterized by a real tilt s. It is presented as a generalization of the natural logarithm, obtained by replacing exponential growth with a power-law form in a controlled way. For a real parameter s, the logaritmilt of x with respect to s is defined by logaritmilt_s(x) = ∫_1^x t^(s−1) dt. This integral evaluates to (x^s − 1)/s when s ≠ 0, and by continuity logaritmilt_0(x) = ln x.

The transform is strictly increasing for x > 0 for all real s, and its inverse is logaritmilt_s^−1(y) = (s y + 1)^(1/s) for s ≠ 0, and e^y for s = 0. The derivative with respect to x is d/dx logaritmilt_s(x) = x^(s−1), with second derivative (s−1) x^(s−2). These relations show that logaritmilt_s behaves like a logarithm when s is near zero, but can vary in curvature as s moves away from zero.

Key properties include domain and range (x > 0 maps to all real numbers), pointwise convergence to the natural logarithm as s → 0, and the explicit dependence on s in both the function and its inverse. In applications within this hypothetical framework, logaritmilt is used to study how changing growth rates influence logarithmic behavior, and it serves as a pedagogical bridge between power functions and logarithms.

History and usage notes indicate logaritmilt is a constructed concept found in teaching examples and speculative discussions, rather than a standard mathematical tool. See also: natural logarithm, power functions, generalized logarithms.