Logarithm base 1

Written by So8res last updated

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, it would send 1 to 0 (because for every ), and it would also send 1 to 1 (because for every ), which demonstrates some of the difficulties with In fact, it would need to send 1 to every number, because and so on. And it would need to send every to , and every to and 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.

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