Can I infer that if I'm told $f(x)=\ln x$ then $f(y)=\ln y$

Question

I have a lengthy economics question, I am told that $f(x)=\ln x$ (the natural log of $x$) but I am also given the functions $f(y)$ to work with. Since $f(.)$ is a function which applies a transformation, am I able to imply that $f(y)=\ln y$ from the transformation $f(x)$ even thought I am not told this?

Answer 1

Yes you can, provided $y$ lies on the domain of the $\ln$ function, which is $(0,\infty)$.

Answer 2

Yes, because when we write 'let $f(y)=\ln y$', this is simply an abbreviation of the statement 'let $f:\mathbb{R^+}\mapsto\mathbb{R}$ be the function defined by $f(y)=\ln y$ for all $y$'. The qualifying statement 'for all $y'$ tells us that $y$ is a dummy variable—the only purpose it is serving is to tell us that each number is mapped to its natural logarithm. Hence, it makes no difference to write 'let $f(x)=\ln x$'.