proposition 7.61 The logarithm

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:1386 · p. 223

Rests on

depends_on corollary 7.36 cor:ana-mvt-consequences ¶
- depends_on theorem 7.35 Mean value theorem ¶
  - depends_on proposition 7.29 Linearity ¶
    - depends_on definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof ch:05-real-analysis@proof-20 ¶
depends_on corollary A.46 Monotone convergence ¶
- depends_on theorem A.40 Least-upper-bound property ¶
  - depends_on definition A.36 Cut ¶
  - depends_on definition A.38 Order on $\R$ ¶
    - depends_on definition A.36 Cut ¶ ↺
  - proves proof app:A-long-proofs@proof-29 ¶
- proves proof app:A-long-proofs@proof-33 ¶
depends_on equation 7.42 eq:ana-exp-derivative ¶
depends_on lemma 7.60 Functional equation of the exponential ¶
- depends_on corollary A.46 Monotone convergence ¶ ↺
- depends_on definition 7.59 The real exponential ¶
  - depends_on definition 7.45 Series ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.4 Convergence ¶ ↺
  - depends_on proposition 7.48 Ratio test ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - depends_on proposition 7.46 Geometric series ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - proves proof ch:05-real-analysis@proof-27 ¶
    - depends_on theorem 7.8 Cauchy criterion ¶ ↺
    - proves proof ch:05-real-analysis@proof-29 ¶
  - depends_on theorem 7.50 Power series; radius of convergence ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
    - depends_on proposition 7.46 Geometric series ¶ ↺
    - proves proof ch:05-real-analysis@proof-31 ¶
- depends_on equation 5.3 eq:lin-binomial ¶
- depends_on proposition 7.49 Cauchy product ¶
  - depends_on corollary A.46 Monotone convergence ¶ ↺
  - depends_on definition 7.45 Series ¶ ↺
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - proves proof ch:05-real-analysis@proof-30 ¶
- proves proof ch:05-real-analysis@proof-38 ¶
depends_on proposition 7.32 Derivative of the inverse function ¶
- depends_on definition 7.14 Inverse function ¶
  - depends_on definition 7.13 Composite function ¶
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on definition 3.47 Bijective map ¶
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- depends_on theorem 7.23 Intermediate value theorem ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - proves proof ch:05-real-analysis@proof-8 ¶
- proves proof ch:05-real-analysis@proof-16 ¶
depends_on proposition 7.27 Differentiable implies continuous ¶
- depends_on definition 7.20 Continuity at a point ¶ ↺
- depends_on definition 7.26 Derivative of a function at a point ¶ ↺
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- proves proof ch:05-real-analysis@proof-11 ¶
depends_on theorem 7.23 Intermediate value theorem ¶ ↺
proves proof ch:05-real-analysis@proof-39 ¶

Supports

depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
- depends_on remark 7.70 What the condition says ¶
depends_on definition 7.62 Real powers ¶
- depends_on definition 7.90 The golden ratio ¶
  - depends_on proposition 7.91 The golden ratio ¶
    - depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
      - depends_on remark 7.95 Where $\varphi$ is used ¶
    - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 125.3 What the theorem does and does not forbid ¶
    - depends_on remark 7.95 Where $\varphi$ is used ¶ ↺
    - depends_on remark 7.94 Three numbers, two kinds ¶
- depends_on proposition 7.63 Laws of real powers ¶
  - depends_on definition 7.90 The golden ratio ¶ ↺
  - depends_on lemma 7.67 The limit over real exponents ¶
    - depends_on proposition 7.68 The natural base ¶
      - depends_on remark 7.70 What the condition says ¶ ↺
  - depends_on proposition 7.68 The natural base ¶ ↺
  - depends_on proposition 7.91 The golden ratio ¶ ↺
  - depends_on proposition 7.89 Irrationality of square roots ¶
    - depends_on proposition 7.91 The golden ratio ¶ ↺
- depends_on proposition 7.89 Irrationality of square roots ¶ ↺
depends_on lemma 7.67 The limit over real exponents ¶ ↺
depends_on proposition 7.68 The natural base ¶ ↺
depends_on proposition 7.63 Laws of real powers ¶ ↺

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Edges

type	direction	node	provenance	where
`depends_on`	→	cor:ana-mvt-consequences	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	→	Monotone convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	→	eq:ana-exp-derivative	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	→	Functional equation of the exponential	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	→	Derivative of the inverse function	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	→	Differentiable implies continuous	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	→	Intermediate value theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	←	$\ee$ is the unique self-reproducing base	declared	parts/02-mathematical-methods/05-real-analysis.tex:1672
`depends_on`	←	Real powers	declared	parts/02-mathematical-methods/05-real-analysis.tex:1433
`depends_on`	←	The limit over real exponents	declared	parts/02-mathematical-methods/05-real-analysis.tex:1563
`depends_on`	←	The natural base	declared	parts/02-mathematical-methods/05-real-analysis.tex:1628
`depends_on`	←	Laws of real powers	declared	parts/02-mathematical-methods/05-real-analysis.tex:1449
`proves`	←	ch:05-real-analysis@proof-39	declared	parts/02-mathematical-methods/05-real-analysis.tex:1402