ln9

ln9 denotes the natural logarithm of 9, that is, the logarithm of 9 with base e. It is the unique real number x satisfying e^x = 9. Using the exponent rule, ln(9) = ln(3^2) = 2 ln(3). Numerically, ln(9) ≈ 2.1972245773362196.

As with other natural logarithms, the domain of ln x is x > 0, and ln x is a strictly increasing function. Its derivative is 1/x, so the derivative at x = 9 is 1/9. The integral ∫ 1/x dx equals ln|x| + C. Logarithms in different bases relate by ln(9) = log10(9) × ln(10). The identity ln(9) = 2 ln(3) also reflects the property ln(a^b) = b ln a.

In mathematical theory, ln(9) is transcendental, a consequence of the Lindemann–Weierstrass theorem, since 9 is a nonzero algebraic number not equal to 1. This places ln(9) among numbers whose nature cannot be expressed as a root of any nonzero polynomial with rational coefficients. ln(9) commonly appears in computations involving growth models, entropy, and solving equations where natural logarithms are natural to the context.