Since the other summer camps are still going on, I can't give you solutions.  Here are some pointers to get you started.

The set of problems focus on logarithm.  Being more familiar with the rules of logarithm (given in the pages prior to the problems) can help you solve these problems; also, working on these problems can help you learn the rules as well.

Q1. Prove that $x^{\log_a y} = y^{\log_a x}$. 

You may try taking the log of both sides and see if they are equal.  It is convenient to take log base $a$ (usually, when taking logarithm or make a base change, any valid base can be used, but some particular base can work better than other ones).

Q2. Assume function $f(x) = \log_2(x^2 + ax + 1)$ is well-defined on all $x\in\mathbb{R}$.  Find the range of possible values of $a$.

This is a domain-related problem.  SInce $f(x)$ should be defined for all $x$, it is required by the logarithm function that $x^2 + ax + 1 > 0$ for all $x$.  Now the problem becomes a quadratic inequality.  The parabola $y=x^2+ax+1$ opens upward, and if the entire parabola is above the $x$-axis, it means there is no real root for the equation $x^2+ax+1=0$, so the discriminant is negative.  You can solve the range of $a$ from there.

Q5. Let $a>0$ and $a\neq 1$, Show that $\displaystyle\log_{a^2} X = \frac{3}{2}\log_{a^3} X$ for all $X>0$.

Try the base-change formula on both sides.  It is convenient to change the base to $a$.

Q7. Evaluate
\[
\frac{5}{\log_2 2016^3} + \frac{2}{\log_3 2016^3} + \frac{1}{\log_7 2016^3}.
\]

This is a base-change question.  Consider the formula $\log_a b = \dfrac{1}{\log_b a}$.

Q8. Given that $1<a<b<a^2$, arrange the following four numbers in increasing order:
\[
2, \quad \log_a b, \quad \log_b a, \quad \log_{ab}a^2.
\]

Compare each value with $1$ first.