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Zermelo-Fraenkel set theory
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Real Riemann integral
Standard natural real logarithm function
Formulation 0
The standard natural real logarithm function is the D4364: Real function \begin{equation} \log : (0, \infty) \to \mathbb{R}, \quad \log(x) = \int^x_1 \frac{1}{t} \, d t \end{equation}
Formulation 1
The standard natural real logarithm function is the D4364: Real function \begin{equation} \log : (0, \infty) \to \mathbb{R}, \quad \log(x) = \int^x_1 \frac{dt}{t} \end{equation}
Formulation 2
The standard natural real logarithm function is the D4364: Real function \begin{equation} \log : (0, \infty) \to \mathbb{R}, \quad \log(x) = \int^x_1 t^{-1} \, d t \end{equation}
Conventions
Convention 0 (Notation for standard natural basic real logarithm function) : The notation used for the D865: Standard natural real logarithm function is $x \mapsto \log(x)$ or $x \mapsto \ln(x)$.
Child definitions
» D5705: Natural real entropy function
» D866: Standard real logarithm function
» D5754: Standard real log-sum-exp function
Results
» R4283: Standard natural basic real logarithm function escapes to positive infinity at positive infinity
» R4284: Standard natural basic real logarithm function escapes to negative infinity at zero
» R4286: Standard natural basic real logarithm function maps one to zero
» R2719: Affine majorant for the standard natural logarithm
» R2658: Standard natural real logarithm function grows linearly near 1
» R4829: Base inversion property of standard natural logarithm function
» R3498: Homomorphism property of the standard natural logarithm
» R4834: