How do you solve #log_(4) (x^2 - 4) - log_(4) (x + 2) = 2#?
For starters, let's determine the set of real numbers where this equation makes sense. It's necessary because we might engage in non-invariant transformations of this equation, which might produce extra solutions or we might lose certain solutions in the course of transformation. Even if we will use only invariant transformations, it's still a good practice to determine what kind of solutions our equation allows.
Any logarithm with a base 4 (as well as with any other positive base) is defined only for positive arguments. That why we have restrictions:
The first condition is equivalent to
The second condition is equivalent to
The only condition that satisfies condition (3) AND either (1) or (2) above is
Now let's examine the problem at hand.
Applying this to our equation and using an identity
By definition of a logarithm,
Applied this to our problem, it means that, by definition of logarithms,
Since equation (5) is true, we can write it as