Haskell logbase error

If you don't need Double Floating precision then use Float type instead and it seems fine. Such as logBase 10 (1000 :: Float) would return 3.0 or functionally logBase 10 . (fromInteger :: Integer -> Float) $ 1000 would do the same.

As per your code toInteger seems redundant since truncate :: (Integral b, RealFrac a) => a -> b already does that job. So may simply do like

len :: Integer -> Integer
len = (+1) . truncate . logBase 10 . (fromIntegral :: Integer -> Float)

This will work correctly up until 9999987.

There is an integer log base function in GHC modules which has type Integer -> Integer -> Int#.

Example usage:

{-# LANGUAGE MagicHash #-}

import Control.Monad
import GHC.Integer.Logarithms ( integerLogBase# )
import GHC.Exts (Int(..))

main = do
  forM_ [(1::Int)..20] $ \n -> do
    let a = 10^n-1
        la = I# (integerLogBase# 10 a)
        b = 10^n
        lb = I# (integerLogBase# 10 b)
    putStrLn $ show a ++ " -> " ++ show la
    putStrLn $ show b ++ " -> " ++ show lb

Output:

9 -> 0
10 -> 1
99 -> 1
100 -> 2
999 -> 2
1000 -> 3
9999 -> 3
10000 -> 4
99999 -> 4
100000 -> 5
999999 -> 5
1000000 -> 6
9999999 -> 6
...
9999999999999999999 -> 18
10000000000000000000 -> 19
99999999999999999999 -> 19
100000000000000000000 -> 20