remark 7.70 What the condition says

Rests on

• depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
  • depends_on equation 7.47 eq:ana-power-derivative ¶
  • depends_on proposition 7.61 The logarithm ¶
    • 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 theorem 7.34 Rolle ¶
        • 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 $\R$ ¶
        • 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 proposition 7.48 Ratio test ¶
        • depends_on theorem 7.50 Power series; radius of convergence ¶
      • 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 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 ¶
        • 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 definition 7.16 Limit ¶
        • depends_on definition 7.9 Real function ¶
      • 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 ¶
  • proves proof ch:05-real-analysis@proof-45 ¶
• depends_on proposition 7.68 The natural base ¶
  • depends_on equation 7.48 eq:ana-growth-law ¶
  • depends_on lemma 7.67 The limit over real exponents ¶
    • depends_on corollary 7.36 cor:ana-mvt-consequences ¶ ↺
    • depends_on definition 7.19 Infinite limits ¶
      • depends_on definition 7.2 Absolute value ¶
      • depends_on definition 7.16 Limit ¶ ↺
    • depends_on lemma 7.65 The integer part ¶
      • depends_on axiom 3.33 Peano axioms ¶
      • depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
        • depends_on definition A.36 Cut ¶ ↺
        • depends_on theorem A.40 Least-upper-bound property ¶ ↺
        • proves proof app:A-long-proofs@proof-32 ¶
      • depends_on lemma 7.64 Well-ordering of $\N$ ¶
        • depends_on axiom 3.33 Peano axioms ¶ ↺
        • depends_on axiom 3.34 Principle of induction ¶
        • proves proof ch:05-real-analysis@proof-41 ¶
      • proves proof ch:05-real-analysis@proof-42 ¶
    • depends_on proposition 7.57 The limit definition ¶
      • depends_on definition 7.53 $\ee$ ¶
        • depends_on definition 7.45 Series ¶ ↺
        • depends_on proposition 7.48 Ratio test ¶ ↺
      • depends_on equation 5.3 eq:lin-binomial ¶ ↺
      • depends_on proposition 7.6 Algebra of limits ¶ ↺
      • proves proof ch:05-real-analysis@proof-37 ¶
    • depends_on proposition 7.61 The logarithm ¶ ↺
    • depends_on proposition 7.63 Laws of real powers ¶
      • depends_on definition 7.62 Real powers ¶
        • depends_on proposition 7.61 The logarithm ¶ ↺
      • depends_on equation 7.42 eq:ana-exp-derivative ¶ ↺
      • depends_on lemma 7.60 Functional equation of the exponential ¶ ↺
      • depends_on proposition 7.31 Chain rule ¶
        • depends_on definition 7.13 Composite function ¶ ↺
        • depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
        • depends_on proposition 7.6 Algebra of limits ¶ ↺
        • proves proof ch:05-real-analysis@proof-15 ¶
      • depends_on proposition 7.61 The logarithm ¶ ↺
      • proves proof ch:05-real-analysis@proof-40 ¶
    • depends_on proposition 7.6 Algebra of limits ¶ ↺
    • proves proof ch:05-real-analysis@proof-43 ¶
  • depends_on lemma 7.60 Functional equation of the exponential ¶ ↺
  • depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
  • depends_on proposition 7.61 The logarithm ¶ ↺
  • depends_on proposition 7.63 Laws of real powers ¶ ↺
  • depends_on proposition 7.6 Algebra of limits ¶ ↺
  • proves proof ch:05-real-analysis@proof-44 ¶
• depends_on remark 7.58 An identity, not a recipe ¶
  • depends_on equation 7.38 eq:ana-e-binomial ¶
  • depends_on lemma 7.54 Truncation error ¶
    • depends_on equation 7.31 eq:ana-e-series ¶
    • 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 ¶
    • proves proof ch:05-real-analysis@proof-34 ¶
  • depends_on proposition 7.57 The limit definition ¶ ↺

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