What do log-equivalent and log-complete mean? I'm reading the paper The Complexity of Satisfiability Problems by Thomas Schaefer(1978). In the paper, he mentions the phrases "log-equivalent" and "log-complete." Searching through the Google results does not seem to give me anything related to complexity theory. I'm guessing it has to do with log-space.
As an example, let $SAT_C(S)$ be the satisfiability problem with constants allowed.  The following are statements in the paper:
"$SAT_C(S)$ is log-equivalent to the graph reducibility problem."
"$SAT_C(S)$ is log-complete in P"
"$SAT_C(S)$ is log-complete in NP."
 A: I hope that you didn't give up and found an answer in appendix of that paper, but since your question is the only related google's result I think that it's worth to put some definitions here.
Let's start with this two definitons:
A function $f$ is log-space computable if there is a deterministic Turing machine which on input $w \in \Sigma^*$ outputs $f(w)$ and uses at most $O(log |w|)$ memory. It's worth noticing that since there is at most $|\Sigma|^{O(log |w|)}$ different states of tape, then such machine works in polynomial time.
For languages (problems) $A, B \subseteq \Sigma^{*}$, A is log-space reducible to $B$ (denoted by $A \leq_{log} B$) iff there is a log-space computable function f such that for all $w \in \Sigma^{*}$, $w \in A$ iff $f(w) \in B$. Observation from previous definition implies that, problem $A$ is also polynomial-time reducible to problem $B$.
And back to phrases from you question:
Problems $A$ and $B$ are log-equivalent  iff $A$ is log-space reducible to $B$ and vice versa ($A \leq_{log} B \land B \leq_{log} A$).
Problem $A$ is log-complete in class of problems $\mathcal{C}$ iff $A$ is an element of $\mathcal{C}$ and for each problem $B$ from $\mathcal{C}$, $B$ is log-space reducible to $A$ ($A \in \mathcal{C} \land \underset{B\in \mathcal{C}}{\forall} B \leq_{log} A$).
