Simplify $x^\frac{1}{(\log_a x)}$ Simplify $x^\frac{1}{(\log_a x)}$
The solution in my textbook is the following:

Since $\log_a (x^\frac{1}{(\log_a x)}) = \frac{1}{\log_a x}$ $\log_a x
 = 1$,
therefore $x^\frac{1}{(\log_a x)} = a^1 = a.$

I understand the law used in the first line is $\log_a (x^y) = y \log_a x$. I also understand that $\log_a x$ multiplied by it's reciprocal equals to 1. However, I do not understand why it makes sense to take the $\log_a$ of our expression in the first place and neither do I understand how the conclusion in the first line helps us arrive at the conclusion in the second line that $x^\frac{1}{(\log_a x)} = a^1 = a.$

First, the expression has sense only for $x > 0$.
So for $x > 0$, the number $y= x^{\frac{1}{\log_a(x)}}$ exists and is $> 0$, so you can compute its $\log_a$. The computation shows that $\log_a(y)=1$.
Now, do you know a lot of numbers whose $\log_a$ are equal to $1$ ?
