How would you go about evaluating y in terms of x in the equation x[sup]y[/sup] = y[sup]x[/sup]? I know that y = x and y = x[sup]x[/sup] both satisfy this equation.
Ignore this thread, I’ve just seen my mistake, y = x[sup]x[/sup] doesn’t satisfy the equation (missing brackets).
The answer’s y = x.
There are other answers as well. y = 2 and x = 4 is a valid solution.
I don’t think that the general solution can be derived algebraically. There is a function, Lambert’s W function, that can be used to get other answers, but I don’t know much about it.
Exponents can be some of the tougher math problems.
A typical approach would be to use logarithms of both sides:
ln (x[sup]y[/sup]) = ln (y[sup]x[/sup])
which is
y ln x = x ln y
- or *
(ln x) / x = (ln y) / y
I’m not quite sure where to go with it from there, though. As you’ve noted while I’ve been previewing my formatting, y=x is trivially a solution. I also note that the function (ln x) / x peaks at x = e, so that there are some values (x > e) that will match those of (y < e), so there should be some other solutions. Solving for them is beyond me.
I’ll add that any general solution should be a mapping from the interval (1,e) to the interval (e, infinity).
Avtually your right, y=x is probably not the only solution, it’s jsut that my other solution y = x[sup]x[/sup] is incorrect.
Here you go: the general solution, algebraically derived.