Is $(H,*)$ a subgroup? I have a few problems like this and am not sure how to get started.  If I can see how to do one, I'm sure I can figure them all out.  Here's the problem.
We have $G = ( \mathbb{R}, +)$ and $H = \{$ log $a: a \in \mathbb{Q}, a >0\}$.   Is $H$ a subgroup of $G$?
I know that I need to prove that $i^{-1} + j \in H$ but I'm not sure how, for $i,j\in H$.   I guess I'm not sure what $i^{-1}$ would be in $H$.  $i=$ (log $a)^{-1}$?

You're considering inverses under the inherited group operation. Here, the inverse is the additive inverse, so what you're asking for is $-\log a + \log b \in H$ for any $\log a, \log b \in H$ (along with $H$ nonempty of course, but this is obvious).
But if $\log a, \log b \in H$, we know that $a, b \in \mathbb{Q}$ and $a, b > 0$. For $-\log a + \log b$ to then be in $H$ is, of course, the same as asking for $\log (a^{-1}b) \in H$, which is the same as asking for $a^{-1}b \in \mathbb{Q}$ and $a^{-1}b > 0$. Given our choice of $a$ and $b$, this is true.
