There is no logarithm base 1, because no matter how many times you multiply 1 by 1, you get 1. If there were a log base 1, though, it would send to (because for every ), and it would also send to (because for every ; this demonstrates some of the difficulties with ), and it would also send to every number (because and so on). It would also send every to , and every to .
If you really want a logarithm base , you can define to be a multifunction with domain and codomain On the input it outputs . On every input it outputs . On every input it outputs . This multifunction can be made to satisfy all the basic properties of the logarithm, if you interpret as , as the interval , and as the interval . For example, , , and . , and . This is not necessarily the best idea ever, but it does give some intuition for how the final form of the logarithm is a multifunction. See also Log_is_a_multifunction.
While you're thinking about weird logarithms, see also Log_base_infinity.