What is the behavior of these functions linear or logarithmic or neither? Please consider the following functions $F$ and $G$. 
\begin{align*}
F(K) =  \log_2 \left(\frac{( 2\sqrt{K-1}+K-2)^2}{(\sqrt{K-1}-1)^2}\right)+(K-1) \log_2(2)
\end{align*}
and the function
\begin{align*}
G(K)=
\log \left( \bigg( \frac{K^2-2}{(K-1)^2}\bigg) \ \frac{(2\sqrt{K-1}+K-2)^2}{(\sqrt{K-1}-1)^2} \right) 
+(K-2)\log_{2}\left(\frac{(K^2-2)}{2(K-1)^2} \frac{(\sqrt{K-1}+1)^2}{(\sqrt{K-1}-1)^2}\right)
+(K-2)\log(2)
\end{align*}
I should note that $K$ is a positive integer greater than 2.
My question is I don't really understand how these functions are behaving a a function of $K$? Are these functions linear or logarithmic as function of $K$ or something else...
Any help would be much appreciated.

First of all,
linear grows faster then logarithmic,
so linear+log gives linear.
Second,
log of a polynomial
or any function that
grows like
$x^n$ for some fixed $n$
is still log
(with a constant),
since
$\log(x^n)
=n \log(x)
$.
OK, now we can begin.
For $F(K)$,
the first term is the log
of a polynomial function of $K$
(it doesn't matter what the degree is),
so it is log.
The second term
is linear times a constant,
so it it linear.
Their sum is
linear+log, which is linear.
For $G(K)$,
the first term is,
as before,
log.
The third term is,
as before linear.
The second term is tricky.
The expression inside the log
is of the form
$\dfrac{K^2+c_1}{2K^2+c_2K+c_3}
$
times,
when you expand it,
$\dfrac{K+c_4\sqrt{K}+c_5}{K+c_6\sqrt{K}+c_7}
$
where the
$c_i$ are constants.
The first term tends to
$\frac12$ for large $K$,
and the second term tends to
$1$ for large $K$,
so their product tends to
$\frac12$ for large $K$.
Therefore the term inside the log tends to $\frac12$,
so the log tends to
$\log(\frac12)$,
which is constant.
Therefore,
the second term is linear.
Therefore
$G(K)$
is linear.
To make these types of problems
easier,
look up "orders of magnitude"
and "big-oh" and "little-oh"
operations.
