logb1x

logb1x is a mathematical function representing the logarithm of x with base b. It is defined as the power to which the base b must be raised to obtain the value x. In other words, if y = logb1x, then b^y = x.

The domain of the logarithmic function logb1x is x > 0, meaning the argument x must be a positive real number. The base b must also be a positive real number and cannot be equal to 1. If b = 1, the function becomes undefined or trivial depending on the context.

Key properties of logarithms include the product rule: logb1(xy) = logb1x + logb1y, the quotient rule: logb1(x/y) = logb1x - logb1y, and the power rule: logb1(x^p) = p * logb1x. These properties are fundamental for simplifying logarithmic expressions and solving logarithmic equations.

Commonly used bases for logarithms are 10 (common logarithm, often denoted as log10x or simply logx) and e (natural logarithm, denoted as lnx). The natural logarithm is particularly important in calculus and many areas of science and engineering.

The graph of y = logb1x is a curve that passes through the point (1, 0) for any valid base b. For bases greater than 1, the function is strictly increasing. For bases between 0 and 1, the function is strictly decreasing. The y-axis (x=0) is a vertical asymptote for the graph of logb1x.