# Is every regular/context free langauge decidable in LogSpace?

I know all the regular languages are decidable but not sure whether it can be done in LogSpace.

It is unknown whether context-free languages can be decided in logarithmic space. The best known result shows that they can be decided in nondeterministic space $O(\log^2 n)$. It is also known that if all context-free languages can be decided in deterministic logarithmic space then $\mathsf{L} = \mathsf{NL}$, which is considered unlikely.
For regular languages, we have $$\text{REG = DSPACE}(O(1)) = \text{NSPACE}(O(1))$$ where REG is the class of regular languages. This can be easily seen from the equivalent formalisms of regular languages. A language is a regular language iff it is the language accepted by a deterministic finite automaton. A language is a regular language iff it is the language accepted by a nondeterministic finite automaton.
In fact, $\text{REG = DSPACE}(o(\log \log n))$. That is, $Ω(\log\log n)$ space is required to recognize any non-regular language [1].