lnQ

lnQ refers to the natural logarithm of a quantity Q, with the natural logarithm denoted by ln. It is defined for positive Q, since the natural logarithm is the inverse of the exponential function e^x on the domain Q > 0. In practical terms, lnQ measures the power to which e must be raised to obtain Q.

Key properties include: for positive numbers a and b, ln(ab) = ln a + ln b, ln(a^k) = k ln a, and ln(a/b) = ln a − ln b. The natural logarithm maps the positive real numbers onto the entire set of real numbers, so its range is (−∞, ∞). The derivative with respect to Q is d/dQ ln Q = 1/Q for Q > 0, and the second derivative is −1/Q^2, indicating concavity downward.

lnQ is related to the exponential function by Q = e^(ln Q) and ln(e^x) = x for all real x. The functions ln and exp are inverse functions of each other on their respective domains.

In applications, lnQ appears in various contexts. In chemistry, for a reaction quotient Q, changes in Gibbs free energy are often written as ΔG = ΔG° + RT ln Q, linking Q to spontaneity under nonstandard conditions. In statistics and information theory, logarithms of likelihoods and probability ratios use natural logs for mathematical convenience, improving additivity and stability in products of probabilities. In mathematics, ln is routinely used in integration, asymptotic analysis, and definitions of special functions.

Common quick examples: ln 2 ≈ 0.6931 and ln(1/2) ≈ −0.6931. Note that ln Q is undefined for Q ≤ 0 unless extended to complex values, where the logarithm becomes multi-valued.