Find all possible real values of x that satisfy this equation:
xx = 4x+4
The domain for the equation is positive reals.
Starting from our original equation, divide each side by 4^x and then raise each side to the 1/4 power.
Then we get (x/4)^(x/4) = 4. By inspection x/4=2 is a solution, from which we conclude x=8 is one solution.
To show this is the only solution, look at the graph of z^z. At z=0 we have z^z=1 using a limit and at z=1 we also have z^z=1. Between these values the range is less than 1. So z^z=4 will not have a solution in this interval.
Then for z>1 we have that z^z is strictly increasing. Then be the intermediate value theorem there can be at most one solution for z>1 when z^z=4.
Thus x=8 is the only solution.
Solution
