Logaritmse

Logaritmse is a fictional mathematical construct used to illustrate an extension of the logarithm. It is not a standard term in mathematics, but may appear in informal expositions or speculative writings. The Logaritmse of a real number x, given a base a and a shift s, is defined as L_{a,s}(x) = log_a(x - s) for x > s, where a > 0 and a ≠ 1. The shift s moves the input threshold, so the logarithm is defined only for x greater than s, and the function satisfies L_{a,s}(a^{t} + s) = t for all real t.

The function is monotone: for a > 1 it is increasing, and for 0 < a < 1 it is decreasing. Its derivative is d/dx L_{a,s}(x) = 1/((x - s) ln a). The asymptotic behavior mirrors the ordinary logarithm: as x grows large, L_{a,s}(x) ~ log_a x, and as x approaches s from above, L_{a,s}(x) → -∞.

Change of base and inverse relationships are straightforward. For any base b > 0, b ≠ 1, L_{a,s}(x) = log_b(x - s)/log_b(a). The exponential family E_{a}(y) = a^y + s serves as the inverse, so L_{a,s}(E_{a}(y)) = y, establishing a direct inverse relationship between the shifted logarithm and its corresponding shifted exponential.

Applications and usage of Logaritmse appear mainly in fictional or illustrative contexts. The shift allows modeling of quantities that become defined only after crossing a threshold or to describe data that are preprocessed by a shift before logarithmic scaling. Generalizations extend to complex values with branch considerations and to higher-dimensional analogues. See also logarithm, change of base formula, and inverse functions.