Logaritmina

Logaritmina, commonly known as the logarithm, is a mathematical function that serves as the inverse of exponentiation. For a base a with a>0 and a≠1, the logarithm of a positive number x is the unique y satisfying a^y = x. It is denoted log_a(x); when the base is e, the function is called the natural logarithm and written ln(x). The base 10 logarithm, often written log10(x) or simply log in many contexts, is widely used in science and engineering.

Core rules: log_a(1) = 0 and log_a(a) = 1. Product rule log_a(xy) = log_a(x) + log_a(y). Power rule log_a(x^r) = r log_a(x). Change of base formula log_a(x) = log_b(x) / log_b(a). Domain x>0; a>0, a≠1; range all real numbers.

Behavior and graphs: for a>1, log_a is increasing; for 0<a<1, it is decreasing. The graph passes through (1,0) and tends to −∞ as x approaches 0+ and to +∞ as x grows without bound. The derivative of log_a(x) is 1/(x ln a) in calculus terms.

History and uses: logarithms were introduced in the early 17th century by John Napier to simplify multiplication and division. Henry Briggs popularized base-10 logs. Today logaritmina are essential in solving exponential equations, data transformation, and in sciences for quantities that span several orders of magnitude. They underpin pH, decibel scales, and various algorithms, including computer science methods for handling multiplicative growth and scale.