# Can a logarithm have a base of 1?

Think about the meaning of the logarithm in a certain base $a$: you have that ${\log}_{a} \left(x\right) = y$ if ${a}^{y} = x$. But if $a = 1$, then ${a}^{y} = 1$ for any $y$. So, you would have chosen a base which can only give $1$ as the result of any power, and so the function would be defined only in one point: the only possible question is "what exponent I must give to $1$ to obtain $1$?" Because for any other number $x$, the question "what exponent I must give to $1$ to obtain $x$?" would be impossible.