For a rational integer $n \in \mathbb{Z}_{+}$, let $\mathfrak{p}(n)$ denote the set of (distinct) prime factors of $n$. Then for a positive constant $c$, let $$f(x) = \vert\{n\in\mathbb{Z}_{+}:\ n\leq x,\textit{ and } \max{\mathfrak{p}(n)} \leq c \log n\}\vert.$$ I.e. $f(x)$ is the number of integers $n$ below $x$ such that there largest prime factor is bounded by $c\log n$. Is the density of such numbers $f(x)/x$ known? Could you refer me to some relevant results on this?
I am aware of some density results on B-smooth numbers but that's not what I'm looking for.
Let $f_u(x)$ denote the number of integers $n \le x$ that only have prime factors at most $x^{1/u}$. If $x$ is big eough, then $x^{1/u} \ge c \log(x) \ge c \log(n)$ for all $n \le x$. So, for big enough $x$,
$$f(x) \le f_u(x).$$
On the other hand, a therorem of Dickman from 1930 says that
$$\lim \frac{f_u(x)}{x} = \rho(u),$$
where $\rho(u)$ can be described fairly explicitly. For example, $\rho(u) = 1 - \log(u)$ for $1 \le u \le 2$. The explicit description of $\rho(u)$ also implies that $\lim_{u \rightarrow \infty} \rho(u) = 0$. In particular, your limit is also zero.
A good reference for these facts (and much more) is here:
https://dms.umontreal.ca/~andrew/PDF/msrire.pdf
(Everything above is contained in the first two pages of that survey, see (1.3) and (1.6))