Alphabeta MathThe Logarithm and General Powers
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 21 results · all verified · 1 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 20 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
The natural logarithm as the inverse of the exponential function
Theorem
Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
Theorem
The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
Theorem
The Euler–Mascheroni constant and the harmonic asymptotic
Definition
Real powers for positive bases, with the zero-base positive-exponent convention
Theorem
The exponential definition of real powers agrees with the existing rational powers
Theorem
The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
Theorem
Continuity and derivatives of positive-base real powers
Definition
The logarithm to a positive base other than one
Theorem
Change of base and inversion of the positive-base real exponential
Definition
Real powers from suprema of rational powers, with the reciprocal convention below base one
Theorem
The rational-supremum construction of real powers agrees with the exponential construction
Corollary
Every rational approximation to a real exponent gives the same limiting real power
Theorem
Landau's root limit: log x is the limit of 2ⁿ times (x^(1/2ⁿ) minus 1)
Theorem
The power series for log(1+x) on (-1,1], including the Abel endpoint
Theorem
Under square summability, the signed product of (1+pₙ) converges iff the series of pₙ converges
Theorem
The logarithm grows more slowly than every positive real power
Theorem
The p-series for a real exponent p converges exactly when p is greater than one
Theorem
The two-point convexity inequality for the exponential function
Theorem
The weighted arithmetic-geometric mean inequality for real weights
Theorem
Young's inequality for conjugate real exponents
Theorem
Holder's inequality for finite sums and conjugate real exponents
Theorem
Minkowski's inequality for finite sums and real exponent p greater than one
Definition
The six hyperbolic functions and their natural domains
Theorem
Addition formulas, identities, parity, and derivatives of the hyperbolic functions
Theorem