Divide both sides by , then we get The left side of the above equation is monotone decreasing while the right side of the equation is monotone increasing.
So, maximum number of solutions is .
Further note that if we consider the original equation and now divide both sides of this equation by then we get,
Then, define
Then ,
But when taking with we get and also
.
So, we can conclude that there exists some such that
(one elegant way to make this claim more rigorous is to use the definition of limit )
So, by the Intermediate Value Property, we get that there is at least one real solution to the equation This shows that the equation has exactly one solution and more precisely this solution is a positive real number when .

In general we can devise a method of solving
when by dividing both sides of this equation by and then the similar argument gives a solution.