Theorems · Theorem · number theory
Nat.le_log_iff_pow_le
∀ {b : ℕ}, 1 < b → ∀ {x y : ℕ}, y ≠ 0 → (x ≤ Nat.log b y ↔ b ^ x ≤ y)(b ^ ·) and log b (almost) form a Galois connection. See also Nat.pow_le_of_le_log and
Nat.le_log_of_pow_le for individual implications under weaker assumptions.
- Defined in
- Mathlib.Data.Nat.Log
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Nat.logstatement · cited by 101
- le_iff_le_iff_lt_iff_ltproof · cited by 25
- Nat.log_lt_iff_lt_powproof · cited by 5
Cited by5
Results whose statement or proof uses this declaration.
- Nat.le_log_of_pow_leproof · cited by 8
- Nat.log_eq_iffproof · cited by 5
- Nat.pow_le_of_le_logproof · cited by 4
- Nat.log2_eq_log_twoproof · cited by 1
- Nat.log_pow_leftproof · cited by 0