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?

There is a general result that if $(x_n){n\in\mathbb{N}}$ is dense and $f(x)$ is continuous, then $\left(f(x_n)\right){n\in\mathbb{N}}$ is dense. — rtybase, Jun 09 '18 at 20:20
Like this one https://math.stackexchange.com/questions/1146931/continuous-function-and-dense-set — rtybase, Jun 09 '18 at 20:27

score 0 · Accepted Answer · answered Jun 09 '18 at 20:29

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.

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