Does there exist a function $f(x)$ so that $$ \lim_{t \to \infty} \frac{\log \,f(e^{tx})}{t} = x^2 ~~~ ? $$
Motivation: The "tropicalization" of a function $f(x_1,\dots,x_n)$ is the function $$ \lim_{t \to \infty} \frac{1}{t} \, \log \, f(e^{tx_1},\dots,e^{tx_n}) $$
This gives the usual "max-plus" tropical algebra: if you take $f(x,y) = x+y$, then this limit is $\mathrm{max}(x,y)$, and if you take $f(x,y) = xy$, then this limit is $x+y$. I'm wondering if it's known if there is any function whose tropicalization is ordinary multiplication?
In other words, is there some $f(x,y)$ so that $\lim \frac{1}{t} \, \log \, f(e^{tx},e^{ty}) = xy$ ? The original question above (in just the single variable $x$) would just be akin to taking $y=x$.
Such a function cannot exist. $$ \lim_{t \to \infty} \frac{\log \,f(e^{tx})}{t} = F(x) $$ for all $x > 0$ implies that $$ F(ax) = \lim_{t \to \infty} \frac{\log \,f(e^{tax})}{t} = a \lim_{t \to \infty} \frac{\log \,f(e^{atx})}{at} = aF(x) $$ for all $a > 0$.
The function $F(x) = x^2$ does not satisfy this homogeneity condition.