Question: Im taking an introduction to logarithms. Of course a short review of exponentiation is inherent for a clear understanding of logarithms.

I was asked to find, for example, $27^x = 3$. (without the use of logarithms) Am i to use pure intuition that 3 is a factor of 27?

Lastly, I was asked to find $2^{2x} - 6 \* 2^{x} + 8 = 0$. It is obvious this is a quadratic equation, however how do i particularly use the formula to solve for the x power.

Answer: 
There is more than one approach.

For your first example, you can either use logarithms as in $$x \log 27 = \log 3$$ or your knowledge that $27=3^3$ so $$3^{3x}=3^1 \text { so } 3x=1$$ or some combination of the two so $$ 3x \log 3= \log 3.$$ 

For the second, $2^{2x}=\left(2^x\right)^2$ so you can turn this into a quadratic you can solve.
For the second equation, take $y=2^x$. Then your equation becomes $$y^2-6y+8=0$$