ILogBase

The Haskell Prelude includes logBase, a floating-point based logarithm-taking function.

Since it is floating-point based, it is limited in number sizes, and has accuracy errors that render it unusable even for simple purposes such as digit counting.

A naive way to implement logBase accurately for Integral types is:

naiveLogBase base
    | base <= 1 = error "base <= 1"
    | otherwise = length . takeWhile (>=base) . iterate (`div` base)

However, this implementation is slow, and has O(result), or O(log(n)) time complexity.

For purposes such as digit counting, an Integral-only logBase can be both accurate and fast:

iLogBase :: Integral a => a -> a -> (a, a)
iLogBase base n
    | base <= 1 = error "iLogBase: base <= 1"
    | n < 0 = error "iLogBase: negative n"
    | n < base = (0, n)
    | otherwise =
        let (res, remain) = iLogBase (base*base) n
            mres = res*2
            (i, r) = if remain < base
                       then (mres, remain)
                       else (mres+1, remain `div` base)
        in (i, r)

The above implementation has O(log(result)), or O(log(log(n))) time complexity.