- From NRICH,
find all real solutions to
Extension: What if x is permitted to be a complex number?
- MathHelp replies,
ab=1 when a=1 or b=0 (and a is nonzero), so the first try at a solution is to solve
and take the union of the results. The first equation gives us and . The second equation gives us 1 and −1. But then we need to throw out because that solution gives us 00, which is undefined. So the results of the first try are:
Noting that (-1)2=1 and that for all integers, k, there are certainly other solutions to ab=1. For all I know at this point, there may be real solutions in x to the pair of equations
where a and b are complex solutions to ab=1.
So I would like to start by characterizing solutions to ab=1. First, the equation is solved whenever a=1, regardless of b. If a is not equal to 1, then,
- ab = eb ln(a) =
and so the complete solution to ab=1 is
The second solution can also be expressed as