I have $0<y<1$. I need to find the domain of $u$ where $u$ is the 1-1 transformation $ u = -2\log{y}$.
Can I perform the following steps? Take Logarithm: $\log{0}<\log{y}<\log{1}$, Multiply by $-2$: $-2\log{0}>-2\log{y}>-2\log{1}$ and obtain $0<u<\infty$?
Otherwise how can I find it? Cause logarithm looks like an increasing function, so if something has an order before, it should also have it after, or is it wrong? In general how do I deal with logarithms or with taking functions to all sides of inequalities?