I need to find all values of the real parameter "a" for which the equation $\left| {a - \log x} \right| + \left| {a + \log x} \right| = 2$ has two solutions. Find the solutions.

Have you tried expanding out the absolute value to take care of all possible combinations?

Hint: a - log(x) + a + log(x) = 2 for a - log(x) > 0 and a + log(x) > 0 is 1 equation. Can you expand the rest and find the two common solutions?

yeah...but that's a dead-end way. What I did was I sqared the equation and so I got rid of one module, and it is much easier to work with only one module than with two, especially having to variables...
If you are interested in the solution:

You might have to re-do it since the post says your equation is too long.