# IF a language L logspace reduces to SAT, does L

If a language L logspace reduces to SAT, does L also reduce to SAT in polynomial time?

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Short answer: Yes, for the same reason that LOGSPACE $\subseteq$ P.
In horrible detail: Let $M$ be a TM that performs a logspace reduction of $L$ to SAT. Say that $M$ has $N$ states and a tape alphabet with $A$ symbols. Since $M$ performs a logspace reduction of $L$, the space taken by $M$ in transforming any element $s$ of $L$ to a SAT instance must be bounded above by $c\cdot\log|s|$ for some constant $c$.
$\def\s#1{\langle T_{#1}, S_{#1}\rangle}$Consider a transformation by machine $M$ of a string $s$ to an instance of SAT. At time $n$, $M$ is in state $S_n$ and its tape is in state $T_n$. There can't be two different times at which both state and tape are the same, or $M$ would be in an infinite loop, which we know doesn't happen. So the running time of $M$ is bounded above by the number of possible (state, tape configuration) pairs, which is $N$ times the total number of possible tape configurations. But at most $c\cdot\log|s|$ tape squares are used, so the total number of tape configurations is at most $A^{c\cdot\log|s|}$, and thus the total time is bounded by $$N\cdot A^{c\cdot\log|s|}.$$ This is a polynomial in $|s|$, so the whole computation is time-bounded by a polynomial in $|s|$, for any $s\in L$, and therefore $L$ is polynomial-time reducible to SAT.