If $f(x) = \log_{x^3}\left(\sqrt{x}\right)$, check whether $f$ is one-to-one and onto where $x\in R^+\setminus\{1\}$. Also write the range of $f$.
Alright, if $f(m) = f(n)$ and if we would prove $m=n$ it would be a one-to-one function. Since $f(x)=\ln(\sqrt{x})/\ln(x^3)$ I equated them, I'm not quite sure how I would verify $m=n$ or $m≠n$.
I proved until $m^n=n^m$ so It can't be one-one since $m \neq n$, right? So function is not one-to-one or onto? Is that correct?
Hints: observe that for any $\;1\neq x>0\;$ we have
$$\log_{x^3}\sqrt x=\frac12\log_{x^3}x=\frac12\frac{\log x}{\log x^3}=\frac16$$
Answer now your question.