Logb

Logb, or the logarithm to base b, is the exponent y that satisfies b^y = x. In the real-valued sense, this definition requires the base b to be positive and not equal to 1, and the argument x to be positive. The logarithm is the inverse function of the exponential base b: if y = logb(x), then b^y = x, and conversely logb(b^y) = y.

Base constraints and behavior: If b > 1, logb(x) is increasing in x; if 0 < b < 1, it is decreasing. The function maps x > 0 to all real numbers. Common special cases include loge(x) (natural logarithm) with base e, and log10(x) (common logarithm) with base 10. The change of base formula allows computation with any positive base: logb(x) = ln(x) / ln(b) = logk(x) / logk(b) for any k > 0, k ≠ 1.

Key properties: logb(1) = 0 and logb(b) = 1. It satisfies logb(xy) = logb(x) + logb(y) and logb(x^r) = r logb(x) for any real r. It is also the unique real-valued function making these product and power rules hold with the domain x > 0.

Examples: log2(8) = 3, log10(1000) = 3, log1/2(4) = -2. In many programming languages, logb may be implemented via a general log function using the change-of-base formula. Complex logarithms extend these ideas with branch considerations beyond the real-valued domain.