One thing to notice
big-theta is an equivalence relation,
big-O and little-o are not.
In other words,
if $f(x) = \Theta\bigl(g(x)\bigr)$,
and $g(x) = \Theta\bigl(h(x)\bigr)$,
$f(x) = \Theta\bigl(f(x)\bigr)$,
$g(x) = \Theta\bigl(f(x)\bigr)$,
$f(x) = \Theta\bigl(h(x)\bigr)$.
To show that big-O is not an
we only need one counterexample.
Just note that,
as $x \to \infty$,
$x = O(x^2)$
but $x^2 \not = O(x)$.
A general principle
which you should get comfortable with
if $f(x) = O(g(x))$ then
$g(x)+f(x) = \Theta(g(x))$.
Some particular results you should know
(and know why they are true)
if $a$ and $b$ are positive constants,
$ax^n = \Theta(bx^n)$,
$\log(x^a) = \Theta(\log(x^b))$,
$\log^a(x) = o(x^b)$,
$\log^a(x) \ne \Theta(x^b)$
If you understand these,
the listed problems should be