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?