La pregunta requiere encontrar todos los valores reales de $x$ para los cuales $$\lfloor \ln x\rfloor \gt \ln\lfloor x\rfloor $$ To start off, one could note that $$\lfloor \ln x \rfloor =\begin{cases} 0,& x\in[1,e) \\ 1,& x\in[e,e^2) \\ 2, &x\in [e^2,e^3) \\ 3,& x\in [e^3,e^4) \\ \vdots \end{cases}$ $ y
$$\ln\lfloor x\rfloor =\begin{cases} 0, &x\in[1,2) \\ \ln 2, &x\in [2,3) \\ \ln 3,& x\in[3,4) \\ \ln 4,& x\in[4,5) \\ \vdots \end{cases}$$Although, from here it is not exactly clear to me how I can proceed. It appeas to me that there are infinitely many intervals of $ x$ for which this inequality is true, but how can I find a generalized form of such an interval? E.g. something of the form $ x \ in \ big (f (k), g (k) \ big)$ for $ k \ in \ mathbb N $ ?
