Denseness of $\{$ log$_{10}$(x) (mod 1) $\}_{n}$ and denseness of $\{$ x (mod 10) $\}_{n}$

Question

If $\{ \log_{10}(x_n) \pmod{1}$ $\}_{n}$ is dense in [0,1] then is $\{ x_n \pmod{10} \}_{n}$ dense in [1,10]? What about vice-versa?

Answer 1

This is implied by the continuity of the function $f(x):=10^x$.

For all $\epsilon > 0$ and for all $y \in [1,10]$ there exists a $\delta$ such that $|x-y| < \delta$ implies $|f(x)-f(y)|< \epsilon.$

Now take $x= \log_{10}(x_n)$ for some $n$, which exists by density.