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

• Finite-bit Box–Muller with certified elementary-function evaluators Algorithm
• One-sample natural error in the Gaussian model Corollary
• A power weight fails at both Aₚ endpoints Counterexample
• An almost-Lipschitz vector field has a unique solution through zero but is not locally Lipschitz there Counterexample
• An optimized mean-field law can lose posterior correlation Counterexample
• Hilbert transform is not strong type (1,1) Counterexample
• Identifiability and KL support do not replace testability Counterexample
• Iid strong law fails at infinite absolute mean Counterexample
• Marginal sub-Gaussianity does not justify an anytime confidence sequence Counterexample
• Omitting the start-time discount changes the policy gradient Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• The claimed univariate clean-quantile count exponent is too strong Counterexample
• The complementarity product needs extra regularity Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• The fundamental Hessian is not absolutely locally integrable Counterexample
• Conjugate-form prior for an exponential family Definition
• Differentiable stationary stochastic policies and score vectors Definition
• Fixed-particle finite-temperature density-matrix free energy Definition
• Heat balls and their time slices Definition
• Score and Fisher information Definition
• A 1 over r singularity on the reference triangle Example
• A bootstrapped critic can bias the first actor update Example
• A degenerate GP concentration function does not bound unknown-scale prior mass Example
• A Dirac mass has precisely sufficiently negative Sobolev order Example
• A finite Bernoulli large-deviation tail Example
• A one-observation Bernoulli-logit Fisher-scoring step Example
• A one-parameter density prior meets the logarithmic contraction conditions Example
• A regularized logistic finite sum with an exact minimizer Example
• A sieve can miss the entropy bound while the posterior contracts exactly Example
• A small sensitivity solve error can reverse a two-design ranking Example
• A state baseline can eliminate this estimator's variance Example
• A strictly plurisubharmonic exhaustion of the convex unit ball Example
• An exponential best-error rate transferred to Chebyshev interpolation Example
• An interior Cauchy principal value reduces to a regular integral Example
• Cauchy law and its characteristic function Example
• Complete separation prevents a finite logistic MLE Example
• Endpoint sensor times are D-optimal for an intercept-slope model Example
• Exact CPO cost feasibility can disagree with a sampled line-search cost check Example
• Flux normalization on every centered sphere Example
• Gauss–Jacobi regularization for x to the minus one half times log x Example

…and 163 more results.

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)