logbM

logbM is a mathematical expression representing the logarithm of a number M with respect to a base b. In simpler terms, it asks the question: "To what power must the base b be raised to obtain the number M?"

The expression logbM is defined only when b is a positive real number, b is not equal to 1, and M is a positive real number. The result of logbM is the exponent x such that b^x = M.

Common bases for logarithms include 10 (common logarithm, often written as log M or log10 M) and e, the base of the natural logarithm (natural logarithm, written as ln M or loge M). The base 2 logarithm (written as log2 M) is also frequently used, particularly in computer science.

Logarithms have several important properties. The product rule states that logb(MN) = logbM + logbN. The quotient rule is logb(M/N) = logbM - logbN. The power rule is logb(M^p) = p logbM. The change of base formula allows conversion between different bases: logbM = logcM / logcb, where c is any valid base.

Logarithms are fundamental tools in various fields, including mathematics, science, engineering, and finance, for simplifying calculations, solving exponential equations, and analyzing data. They are used in areas like measuring the intensity of earthquakes (Richter scale), sound (decibels), and acidity (pH scale).