Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm

Depends on

• The natural logarithm as the inverse of the exponential function
• The exponential addition formula $\exp(x+y)=\exp(x)\exp(y)$
• The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$
• The exponential function is strictly increasing
• Continuous inverse theorem: a continuous injective $f$ on an interval $I$ is a bijection onto the order-convex set $f[I]$, and the inverse $g : f[I] \to I$ is continuous and strictly monotone in the same sense as $f$

Used by

• Every continuous f with f(xy)=f(x)+f(y) is f(x)=c log x for a unique c, including c=0 Corollary
• Lyapunov central limit theorem Corollary
• Iid strong law fails at infinite absolute mean Counterexample
• The logarithm is not uniformly continuous on the positive half-line Counterexample
• A continuous argument computed along a spiralling contour Example
• A family containing K₁ is viral for vacuous reasons Example
• A parameter ledger for the high-girth, high-chromatic alteration proof Example
• A positive convex function need not be log-convex Example
• A positive non-log-convex solution of the Gamma functional equation Example
• Comparing the two quantitative density scales Example
• Dropping f(e)=1 leaves the whole family c log, including logarithms to other bases and the zero function Example
• Every hereditary graph class of bounded order has the Erdős–Hajnal property Example
• Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 Example
• The Gauss map preserves Gauss measure Example
• Weighted interval volume Example
• A large Y-part in a structural comb partition yields the clique-or-stable-set outcome Lemma
• A wide integral geometric layer forces the complete-or-anticomplete property-(*) blockade Lemma
• Abel summation recovers the prime-counting function from theta Lemma
• Absolute real powers are Borel measurable and convex Lemma
• Admissible parameters for the density recursion Lemma
• Clarkson inequalities in both exponent ranges Lemma
• Every smaller positive exponent is again an Erdős–Hajnal constant Lemma
• Log-convex solutions of the Gamma recurrence obey the Bohr--Mollerup factorial squeeze Lemma
• Qid logarithmic and constant divisibility Lemma
• The Mercator series, its value at 1 and the product law determine log on all positive reals, while the series alone is only local Lemma
• Alon–Pach–Solymosi: if H₁ and H₂ have the Erdős–Hajnal property, so does the graph obtained from H₁ by substituting H₂ for a vertex 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
• Bertrand's postulate Theorem
• Change of base and inversion of the positive-base real exponential Theorem
• Chebyshev's theta function has linear lower and upper bounds Theorem
• Euler's Beta integral converges exactly for two positive parameters Theorem
• Euler's Gamma integral converges exactly for positive real parameters Theorem
• Every finite family with the Erdős–Hajnal property is viral Theorem
• Every nonempty n-vertex graph satisfies hom(G)≥ 1/2 log₂ n Theorem
• Finite variance logarithmic rate for iid sums Theorem
• For every n≥16 there is an n-vertex graph with hom(G)<3 log₂ n Theorem
• For every t≥1, the class of Kₜ-free graphs has the Erdős–Hajnal property Theorem
• Hadamard three-lines theorem Theorem
• If k≥1 and n≥3k² 2ᵏ, an n-vertex tournament with property Sₖ exists Theorem
• log is the unique continuous f:(0,∞)→ℝ with f(xy)=f(x)+f(y) and f(e)=1 Theorem

…and 16 more results.

Dependency tree · two levels

29 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)