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

• Finite endpoint power substitution followed by Gauss–Jacobi quadrature Algorithm
• Finite-candidate information design Algorithm
• Finite-horizon upper-confidence-bound bandit algorithm Algorithm
• LinUCB with a self-normalized confidence radius Algorithm
• Validation temperature scaling Algorithm
• Variance-aware quantum Monte Carlo Algorithm
• A PAC-Bayes bound for a finite tuning grid Corollary
• A sample-size consequence for compression schemes Corollary
• Compatible extensions from the finite simple core Corollary
• ERM excess risk from Rademacher uniform deviation Corollary
• LinUCB has expected pseudo-regret of order d√n log(nL) Corollary
• One-sample natural error in the Gaussian model Corollary
• Polynomial and exponential minimax estimation rates 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
• A data-dependent prior can invalidate the PAC-Bayes bound Counterexample
• Assuming choice, a Hamel-basis additive map transported through exp gives a discontinuous logarithmic function that is not c log Counterexample
• Disconnected comparison support does not identify Gibbs policy mass Counterexample
• Equal posterior means do not control the full posterior Counterexample
• Fidelity and trace-distance inequalities do not imply QRE data processing Counterexample
• Knapp rules out extension below the Tomas exponent Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Subcritical W^1,p is not closed under multiplication Counterexample
• The claimed univariate clean-quantile count exponent is too strong Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• The logarithm is not uniformly continuous on the positive half-line Counterexample
• Zero reference support makes the DPO reward ratio undefined Counterexample
• Average-cost fixed-length codes and operational DMC capacity Definition
• Causal feedback codes for a finite cost-constrained DMC Definition
• Causal feedback codes for a finite-dimensional Gaussian channel Definition
• Chebyshev's theta function Definition
• Complex logarithms, the principal logarithm, and principal and multivalued complex powers Definition
• Density-defined Jensen–Shannon divergence Definition
• Dirichlet series Definition
• Finite DMC information density Definition
• Finite DMC with a nonnegative input cost and feasible budget Definition
• Finite KL-regularized policy improvement Definition
• Finite Kullback–Leibler divergence Definition
• Finite Rényi divergence Definition

…and 262 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)