log_2_in_interval

formal source

 336  valid := by norm_num
 337
 338/-- log(2) is contained in log2Interval - PROVEN using Mathlib's log_two bounds -/
 339theorem log_2_in_interval : log2Interval.contains (log 2) := by
 340  simp only [contains, log2Interval]
 341  constructor
 342  · -- 0.693 ≤ log 2
 343    have h := Real.log_two_gt_d9  -- 0.6931471803 < log 2
 344    have h1 : ((693 / 1000 : ℚ) : ℝ) < log 2 := by
 345      calc ((693 / 1000 : ℚ) : ℝ) = (0.693 : ℝ) := by norm_num
 346        _ < (0.6931471803 : ℝ) := by norm_num
 347        _ < log 2 := h
 348    exact le_of_lt h1
 349  · -- log 2 ≤ 0.694
 350    have h := Real.log_two_lt_d9  -- log 2 < 0.6931471808
 351    have h1 : log 2 < ((694 / 1000 : ℚ) : ℝ) := by
 352      calc log 2 < (0.6931471808 : ℝ) := h
 353        _ < (0.694 : ℝ) := by norm_num
 354        _ = ((694 / 1000 : ℚ) : ℝ) := by norm_num
 355    exact le_of_lt h1
 356
 357/-- Interval containing log(10) ≈ 2.3025 -/
 358def log10Interval : Interval where
 359  lo := 230 / 100
 360  hi := 231 / 100
 361  valid := by norm_num
 362
 363/-- log(10) is contained in log10Interval.
 364    Proof using log(10) = log(2) + log(5) and Mathlib bounds.
 365    log(2) ≈ 0.693, log(5) = log(10/2) requires log(10).
 366    Instead: log(10) = 2*log(√10), but √10 computation is circular.
 367    Best approach: log(10) = log(2) + log(5) where log(5) = log(4*5/4) = 2*log(2) + log(1.25)
 368    So log(10) = 3*log(2) + log(1.25) -/
 369theorem log_10_in_interval : log10Interval.contains (log 10) := by