Regular languages under change of encoding Consider a regular language $L$ with alphabet $\Sigma = \{0,1\}$.
Can we say that the set of strings in $L$ (representing non-negative integers in binary encoding) when represented in some other encoding (say unary, or, decimal) also form a regular language ?

Formally, is it the case that the language $L' = \{w | w \in 1^*, \exists w' \in L \text{ such that the unary encoding of } w' \text{ is } w \}$ also regular? 

What happens when the base is shifted to a lower number? For example can we also say anything about the following language:
$L'' = \{w | w \in \{0,1,2\}^*, \exists w' \in L \text{ such that the ternary encoding of } w' \text{ is } w \}$
The answer is no.
I'll give an example of a language $L$ which is regular in binary but not in unary:

Consider $L=\{10^k|k\in \mathbb{N}\}$. The corresponding language in unary is $L'=\{1^{2^k}|k\in \mathbb{N}\}$.

It's easy to see that $L$ is regular while $L'$ is not even context free.

L'' also isn't regular either, by the link @Sylvain posted in his comment.