logaritmet

Logaritmet is a term used to refer to the logarithm, a mathematical function that expresses the exponent required to reach a given number from a fixed base. For a base b>0 and b≠1, the logarithm of x>0 is written log_b(x). The natural logarithm uses base e and is commonly written ln x; base 10 logarithms are called common logarithms and written log_10 x; base 2 logarithms are used in computer science as binary logarithms log_2 x.

Key properties include log_b(xy)=log_b(x)+log_b(y), log_b(x^k)=k log_b(x), and the change of base formula log_b x = log_k x / log_k b for any positive k≠1. The logarithm is the inverse function of the exponential b^x, giving log_b(b^x)=x and b^{log_b x}=x.

Computation historically relied on logarithm tables and rules of thumb; today it is performed by calculators and software. The short series ln(1+x)=x−x^2/2+... supports analytic work near x=0.

Common bases and their uses: base 10 (engineering and science), base e (calculus and analysis), base 2 (computer science). Logarithms linearize exponential growth, aid data compression, and appear in scales such as decibels and the Richter scale, as well as in pH calculations.

History: logarithms were introduced in the early 17th century by John Napier, with later developments by Henry Briggs and followers, leading to widespread computational tables. See also exponential function; natural logarithm; change of base formula.

Note about term: In some languages, logaritmet is a variant spelling or regional term for logarithm. The article treats logaritmet as synonymous with logarithm.