Logarithm base 1

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 ${\log }_b\left(1\right)=0$ for every $b$), and it would also send $1$ to $1$ (because ${\log }_b\left(b\right)=1$ for every $b$; this demonstrates some of the difficulties with ${\log }_1$), and it would also send $1$ to every number (because ${\log }_1(1\cdot 1)$ $={\log }_1\left(1\right)+{\log }_1\left(1\right)=2,$ and so on). It would also send every $x>1$ to $\infty$, and every $0<x<1$ to $-\infty$.

If you really want a logarithm base $1$, you can define ${\log }_1$ to be a multifunction with domain $R^{+}$ and codomain $R\cup \{\infty ,-\infty \}.$ On the input $1$ it outputs $R$. On every input $x>1$ it outputs $\left\{\infty \right\}$. On every input $0<x<1$ it outputs $\{-\infty \}$. This multifunction can be made to satisfy all the basic properties of the logarithm, if you interpret $=$ as $\in$, $1^{\left\{\infty \right\}}$ as the interval $(1,\infty )$, and $1^{\{-\infty \}}$ as the interval $(0,1)$. For example, $0\in {\log }_1\left(1\right)$, $1\in {\log }_1\left(1\right)$, and ${\log }_1\left(1\right)+{\log }_1\left(1\right)\in {\log }_1(1\cdot 1)$. $7\in lo{g}_1\left(1^7\right)$, and $15\in 1^{{\log }_1\left(15\right)}$. 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.