lnM

lnM is commonly used to denote the natural logarithm of a quantity M, that is, the logarithm with base e. In standard real-valued contexts, M is a positive real number, and ln M serves as the inverse function of the exponential e^x on the real line. The notation can also appear in more advanced settings where M is a matrix or other object, in which case lnM refers to a related but distinct concept.

For real numbers, ln M is defined for M > 0. Key properties include: continuity and differentiability with derivative d/dM ln M = 1/M; the identities ln(AB) = ln A + ln B, ln(A^k) = k ln A, and ln(A/B) = ln A − ln B (for positive A and B); and the exponential relation e^{ln M} = M and ln(e^x) = x for all real x. The function is strictly increasing, with ln 1 = 0, and its range is the entire real line.

In linear algebra, ln(M) may denote the matrix logarithm of a square invertible matrix M. A matrix X satisfies e^X = M. There can be multiple matrix logarithms, and a logarithm always exists over the complex field for invertible M; the real case is more delicate and not always possible. Practically, logarithms of matrices are computed via decompositions such as the Jordan form or the Schur decomposition, often using scaling-and-squaring methods combined with Padé approximants.

Examples illustrate the concepts: for a positive real M, ln M is the real number whose exponent equals M; for example, ln 4 ≈ 1.386. In matrix contexts, if M = e^A for some matrix A, then ln M = A, up to choices arising from the non-uniqueness of the logarithm.

---