exp and ln: the two halves of eml

The EML operator is built directly from two functions you already know: exp and ln. A quick refresher on their identities will make every derivation in the paper feel natural.

Definitions

• exp(x) = e^x, the continuous extension of repeated multiplication.
• ln(x) = the inverse of exp. ln(x) is the unique number such that e^(ln x) = x.

Their domain/range:

• exp: ℝ → ℝ⁺ (always positive)
• ln: ℝ⁺ → ℝ (requires positive input)

Key identities

These are the scaffolding of every EML tree:

ln(1)      = 0
ln(e)      = 1
exp(0)     = 1
exp(1)     = e

exp(ln x)  = x
ln(exp x)  = x

exp(a + b) = exp(a) · exp(b)
ln(a · b)  = ln(a) + ln(b)

How the identities become EML

Each identity corresponds to a simplification rule in an EML tree.

• eml(x, 1) = exp(x) − 0 = exp(x) uses ln(1) = 0.
• eml(1, 1) = e − 0 = e uses exp(1) = e and ln(1) = 0.
• The identity tree eml(1, eml(eml(1, eml(x, 1)), 1)) reduces to ln(exp(x)) = x.

Spend 15 minutes making sure the eight identities above feel trivial and the rest of the paper opens up.