Powers

Powers

bⁿ is familiar when n is a whole integer (positive or negative). Here's an algorithm to compute it.

let rec pow (i, u) = 
  if i < 0 then 1.0 / (pow (-i, u))
  else if i = 0 then 1.0 else u * pow ((i-1), u)

It is also familiar to us when c/d = n that is, for n a rational number (then the solution is found by taking the dth root of b^c). Less so though I think when x = n is some arbitrary real. That is, generally speaking how is b^x computed?

Logarithms and exponentiation. Since b = e ^{log b} then b^x = (e^{log b})^x = e ^{x (log b)}.

let powf (b, x) = exp (x * log b)

This only works when b is real and positive. The case negative b leads into the theory of complex powers.

“Sometimes,' said Pooh, 'the smallest things take up the most room in your heart.”