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 1 to 0, because for every . It would also send 1 to 1, because for every . It would also send 1 to every number at once, because and so on; this illustrates some of the difficulties inherent with logarithms base 1. And that's just when the input is 1. When the input is the output would have to be and when the input is the output would have to be Those aren't numbers, so there's no logarithm base 1.
But if you really want a logarithm base , you can define to be a multifunction from to 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 hey, the final form of the logarithm was already a multifunction, so whatever. See also Log_is_a_multifunction.
While you're thinking about weird logarithms, see also Log base infinity.