logarítmicos

Logarítmicos, in Spanish and Portuguese usage, refer to logarithmic concepts in mathematics. A logarithm is the exponent to which a base must be raised to produce a given positive number. For a base b > 0 with b ≠ 1 and x > 0, log_b(x) is the unique number y such that b^y = x. The natural logarithm ln(x) is log base e, and log base 10 is the common logarithm.

Key properties include log_b(xy) = log_b(x) + log_b(y); log_b(x^k) = k log_b(x); and log_b(x/y) = log_b(x) − log_b(y). Change of base: log_b(x) = log_k(x) / log_k(b) for any positive base k ≠ 1. The function log_b(x) is increasing when b > 1 and decreasing when 0 < b < 1; it is the inverse of the exponential function b^x.

Bases and usage: Bases other than e or 10 are common in theory and applications. Logarithms compress wide-ranging data and convert multiplication into addition, aiding computation and analysis. They are widely used in science, engineering, economics, and data visualization, particularly on logarithmic scales. Practical examples include pH (negative log of hydrogen ion concentration), decibel levels, and the Richter scale. In computer science, logarithms describe algorithmic complexity, such as O(log n).

History: Logarithms were introduced in the early 17th century by John Napier and were later developed for base-10 by Henry Briggs, revolutionizing calculation before the advent of electronic calculators. Today, logarithms remain fundamental in mathematics, numerical methods, and data analysis.