Solve the following :
Find all positive real numbers such that
…let us say this is given equality A
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:
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.