log3

Logarithm with base 3, denoted log3(x) or log base 3 of x, is the inverse function of the exponential 3^x. It is defined for positive inputs x > 0; its output is any real number y such that 3^y = x. In other words, log3(x) gives the exponent needed to raise 3 to obtain x.

Notation and change of base: log3(x) can be expressed using other logarithms as log3(x) = ln(x) / ln(3) or log3(x) = log10(x) / log10(3). More generally, for any base b > 0 with b ≠ 1, logb(x) = ln(x) / ln(b) or logk(x) / logk(b). This enables easy calculation with common log or natural log functions found in calculators and programming libraries.

Key properties include log3(1) = 0 and log3(3) = 1; logarithm rules such as log3(xy) = log3(x) + log3(y) and log3(x^k) = k log3(x). The function is increasing on its domain and has derivative d/dx log3(x) = 1 / (x ln(3)).

Examples: log3(9) = 2 because 3^2 = 9, and log3(27) = 3 because 3^3 = 27. For x > 0, log3(x) is positive when x > 1 and negative when 0 < x < 1.

Applications and computation: log3(n) is used in contexts where a base-3 scale is natural, and in algorithm analysis it is often treated as Θ(log n) since changing the base only multiplies the result by a constant factor. In software, log base 3 is typically computed via log(x)/log(3) or ln(x)/ln(3).