I defined $a^b = \sup\{a^q:q\in\mathbb Q, q < b\}$ where $b$ is a real number and $a$ is a real number that is not smaller than $1$. When $ 0 < a < 1$, I defined $a^b$ as the reciprocal of $(1/a)^b$. And I am now proving the exponent laws.
I'm stuck in proving $(ab)^c = a^c \times b^c$ where one of a, b is smaller than 1 and the other is not. How can I proceed? I think this is not a difficult problem, but I can't get an idea.
The case where $a$ and $b$ are both bigger than $1$ or both smaller than $1$ were proved. Also, $a^b \times a^c = a^{b + c}$ and $(a^b)^c = a^{bc}$ were proved, and I can use the fact that exponent laws for rational numbers hold.