I'd like a clarification for this exercise.

$$ 7^x \equiv 19\mod30$$

He says: ''the $19$ has to be written like a power of $7$ otherwise it wouldn't be solvable.''

So he writes 

$$ 7^x \equiv 7^2\mod30$$

*First question:* Why do I have to write $19$ as a power of $7$? And why $7^2$ and not $7^5$ ?

Then he proceeds by writing 

\begin{align} 7^x-2 &\equiv 1\mod30\\
y&=x-2 \\
7^y &\equiv 1\mod30\\
x-2 &\equiv 0\mod30\\
y &\equiv 0\mod4\end{align}

What is the reasoning in general about this procedure? What am I looking for when solving this type of exercise?