1) $$f(n) = 2^n\,,\quad g(n) = (3/2) ^ n$$
Is $f(n) = \Theta(g(n))$? Can someone please explain this to me ?
2)$$f(n) = n^2+\log n\,,\quad g(n) = n^2$$
I know that $f(n) = \Theta(g(n))$ but how can I get the constant $c$ to prove the equation for $\Theta$?
For the second problem, just note that, $\forall n \geq 1$
$$ |f(n)| = |n^2+\log n| < n^2 + n \leq 2 n^2,$$
where the fact that $\log(n) < n $ has been used. For the other inequality, observe that
$$ |f(n)| = |n^2+\log n| \geq n^2, $$
since $\ln(n)\geq 0$ $\forall n\geq 1$.