The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t

Depends on

• The natural logarithm as the inverse of the exponential function
• Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c \in I$ with $f'(c) \ne 0$, then the inverse $g$ is differentiable at $f(c)$ with $g'(f(c)) = 1/f'(c)$; and if $f'(c) = 0$ then $g$ is not differentiable at $f(c)$
• The exponential function is smooth and $(\exp)'=\exp$
• The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F'(c) = f(c)$; in particular a continuous $f$ has $F$ as a primitive
• The integral with oriented limits: $\int_a^a f := 0$ and $\int_b^a f := -\int_a^b f$
• Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
• A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant

Used by

• An almost-Lipschitz vector field has a unique solution through zero but is not locally Lipschitz there Counterexample
• Iid strong law fails at infinite absolute mean Counterexample
• Cauchy law and its characteristic function Example
• Geodesics in the Poincare upper half-plane Example
• log|z| is harmonic on the punctured plane Example
• The Gauss map preserves Gauss measure Example
• Weighted interval volume Example
• x⁻ᵃ on (0,1) and on (1,∞) calibrates Lᵖ membership Example
• Abel summation recovers the prime-counting function from theta Lemma
• Chernoff bound for independent bernoulli trials Lemma
• Clarkson inequalities in both exponent ranges Lemma
• Qid logarithmic and constant divisibility Lemma
• Stirling's factorial asymptotic holds up to a positive constant Lemma
• The harmonic sum is log x plus gamma plus O(1/x) Lemma
• The summatory logarithm is x log x minus x plus O(log x) Lemma
• Continuity and derivatives of positive-base real powers Theorem
• Euler's Beta integral converges exactly for two positive parameters Theorem
• Euler's Gamma integral converges exactly for positive real parameters Theorem
• Five characterisations of the natural logarithm are equivalent: inverse exponential, integral, continued Mercator series, Landau root limit and the normalised functional equation Theorem
• Mertens' first theorem for primes Theorem
• Mertens' second theorem for primes Theorem
• The Euler–Mascheroni constant and the harmonic asymptotic Theorem
• The p-series for a real exponent p converges exactly when p is greater than one Theorem
• The power series for log(1+x) on (-1,1], including the Abel endpoint Theorem
• The real Gamma function is smooth and its derivatives are logarithmic moments Theorem

Dependency tree · two levels

53 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)