Solve the following :

Find all positive real numbers such that

…let us say this is given equality A

Solution:

Use the following inequality: with equality iff

So, we observe that : ,

Hence, LHS of the given equality is greater than or equal to:

Now, let us consider the RHS of the given equality A:

we have to use the following standard result:

So, applying the above to RHS of A:

.

But, RHS is equal to LHS as given in A:

That is,

Now, just a few steps before we proved that LHS is also greater than or equal to : That is,

The above two inequalities are like the following: and ; so what is the conclusion? The first inequality means or ; clearly it means the only valid solution is .

Using the above brief result, we have here:

Hence, we get , which in turn means that (by applying the definition of absolute value):

, which implies that .

Substituting these values in the given logarithmic absolute value equation, we get:

, that is , and as , this implies that which in turn means also.