qLog :: String -> String -> IO ()
qLog l msg = log' "" msg where
  log' "I" msg = infoM "" msg
  log' "D" msg = debugM "" msg
  log' "E" msg = errorM "" msg
  log' "W" msg = warningM "" msg
  log' _ msg = infoM "" msg

λ> qlog "D" "Thing!"
[DEBUG::2017-01-18 20:38:19 EST] Thing!

λ> qlog "I" "Thing!"
[INFO::2017-01-18 20:38:19 EST] Thing!

λ> qlog "Thing!"
[INFO::2017-01-18 20:38:19 EST] Thing!

qLog2 :: Maybe String -> String -> IO ()
qLog2 l msg = case l msg of
     (Just "D") msg -> debugM "" msg
     (Just "I") msg -> infoM "" msg
     Nothing  msg -> infoM "" msg

qLog3 :: Maybe String -> String -> IO ()
qLog3 l msg
  | Just "D" msg = debugM "" msg
  | Just "I" msg = infoM "" msg
  | Nothing  msg = infoM "" msg

qLog4 "D" msg = debugM "" msg
qLog4 "I" msg = infoM "" msg
qLog4 msg  = infoM "" msg

Equations for ‘qLog4’ have different numbers of arguments

qLog2 :: Maybe String -> String -> IO ()
qLog2 l msg =
  case l of
    Just "D" -> debugM "" msg
    Just "I" -> infoM "" msg
    Nothing  -> infoM "" msg

qLog :: Maybe String -> String -> IO ()
qLog l msg = case l of
    Just "D" -> debugM "" msg
    Just "I" -> infoM "" msg
    -- etc.
    Nothing  -> infoM "" msg

-- Feel free to use less terse names.
data LogLevel = D | I | W | E
    deriving (Eq, Show)

qLog :: Maybe LogLevel -> String -> IO ()
qLog l msg = case l of
    Just D -> debugM "" msg
    Just I -> infoM "" msg
    Just W -> warningM "" msg
    Just E -> errorM "" msg
    Nothing -> infoM "" msg

-- I added a few superflous parentheses to emphasise what is going on.
qLog :: Maybe LogLevel -> String -> IO ()
qLog ml msg = case ml of
     Just l -> (fLog l) "" msg
     Nothing -> infoM "" msg
     where
     fLog :: LogLevel -> (String -> String -> IO ())
     fLog l = case l of
         D -> debugM
         I -> infoM
         W -> warningM
         E -> errorM

GHCi> :t maybe
maybe :: b -> (a -> b) -> Maybe a -> b

-- There are also some superflous parentheses here.
qLog :: Maybe LogLevel -> String -> IO ()
qLog ml msg = (maybe infoM fLog ml) "" msg
     where
     fLog :: LogLevel -> (String -> String -> IO ())
     fLog l = case l of
         D -> debugM
         I -> infoM
         W -> warningM
         E -> errorM