The natural logarithm as the inverse of the exponential function

Depends on

• The exponential is a continuous bijection from $\mathbb{R}$ onto $(0,\infty)$

Used by

• Compatible extensions from the finite simple core Corollary
• The integral logarithm L is the published natural logarithm Corollary
• The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
• The winding number is the increment of a continuous argument divided by 2π Corollary
• Assuming choice, a Hamel-basis additive map transported through exp gives a discontinuous logarithmic function that is not c log Counterexample
• The logarithm is not uniformly continuous on the positive half-line Counterexample
• Chebyshev's theta function Definition
• Complex logarithms, the principal logarithm, and principal and multivalued complex powers Definition
• Dirichlet series Definition
• Log-convex positive functions Definition
• Real powers for positive bases, with the zero-base positive-exponent convention Definition
• The Euler-Mascheroni constant Definition
• The logarithm to a positive base other than one Definition
• The Meissel-Mertens constant Definition
• The Riemann zeta function on the half-plane Res>1 Definition
• The von Mangoldt function Definition
• A continuous argument computed along a spiralling contour Example
• A parameter ledger for the high-girth, high-chromatic alteration proof Example
• Cauchy law and its characteristic function Example
• Every hereditary graph class of bounded order has the Erdős–Hajnal property Example
• Geodesics in the Poincare upper half-plane Example
• log 2 from four characterisations: hyperbolic area, the alternating harmonic series, Landau iterates and exp⁻¹(2) Example
• Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 Example
• Absolute real powers are Borel measurable and convex Lemma
• Chernoff bound for independent bernoulli trials Lemma
• Clarkson inequalities in both exponent ranges Lemma
• Exponential contraction of projection away from a quasiconvex set Lemma
• Finite simple analytic families and their exact endpoint norms Lemma
• The Dirichlet series for zeta converges absolutely and locally uniformly on the half-plane Res>1 Lemma
• The summatory logarithm is x log x minus x plus O(log x) Lemma
• A circle traversed k times has winding number k inside and 0 outside Theorem
• An entire function of polynomial growth is a polynomial Theorem
• An n-vertex graph of minimum degree δ>1 has a dominating set of size at most n(log(δ+1)+1)/(δ+1) Theorem
• Five characterisations of the natural logarithm are equivalent: inverse exponential, integral, continued Mercator series, Landau root limit and the normalised functional equation Theorem
• Hadamard three-lines theorem Theorem
• If k≥1 and n≥3k² 2ᵏ, an n-vertex tournament with property Sₖ exists Theorem
• Morse stability with explicit parameter dependence Theorem
• Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm Theorem
• Ostrowski's theorem for the rationals Theorem
• Standard Maclaurin expansions Theorem

…and 6 more results.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)