EML: The NAND of Mathematics
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transcendental

Natural logarithm · ln(x)

The natural log reassembles itself in three nested EML calls, a clean contrast to exp's single-node form.

depth 3
Domain: x ∈ ℝ⁺
Definition
ln⁡(x) = eml(1, eml(eml(1, x), 1))\ln(x) \ = \ \texttt{eml(1, eml(eml(1, x), 1))}

Formula

eml(1, eml(eml(1, x), 1))
Evaluate
1at x = 2.718281828459045

Reference: ln(2.718281828459045) ≈ 1

EML tree
1111x2.7183eml1.718311eml5.5749eml1

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Step-through evaluation
  1. 1.depth 1const 1
  2. 2.depth 3const 1
  3. 3.depth 3x = 2.718282
  4. 4.depth 2exp(1) − ln(2.718282) = 1.718282
  5. 5.depth 2const 1
  6. 6.depth 1exp(1.718282) − ln(1) = 5.574942
  7. 7.depth 0exp(1) − ln(5.574942) = 1
Derivation
  1. 1.Let a = eml(1, x) = e − ln(x).
  2. 2.Let b = eml(a, 1) = exp(a) − 0 = exp(e − ln(x)).
  3. 3.Then eml(1, b) = e − ln(b) = e − (e − ln(x)) = ln(x).