# ILogBase

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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.