ln10

ln 10 is the natural logarithm of 10, written as ln(10). It is the inverse of the exponential function e^x and is the unique real number x such that e^x = 10.

Numerical value: ln(10) ≈ 2.302585093.

Relation to other logarithms: ln is the base-e logarithm. The common logarithm log10 is related by log10 x = ln x / ln 10. In particular, log10 10 = 1.

Properties: ln(10) is irrational and, in fact, transcendental. By results in transcendental number theory (such as the Lindemann–Weierstrass theorem), e^α is algebraic for α ≠ 0 only when α is transcendental; since e^{ln(10)} = 10 is algebraic and ln(10) ≠ 0, ln(10) must be transcendental.

Occurrence and uses: ln(10) frequently appears in calculations that convert between natural and common logarithms, in formulas involving exponential growth or decay, in integrals and series that involve natural logs, and in various areas of science and engineering where base-e expressions are common. It also serves as a convenient constant in base-change relations and in evaluating expressions that involve e and 10.