# A maths questions **URL:** \ **Category:** Factual Questions\ **Created:** [August 13, 2003, 5:50pm UTC](https://boards.straightdope.com/t/a-maths-questions/194959 "2003-08-13T17:50:57Z")\ **Posts on this page:** 7\ **Page:** 1
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [August 13, 2003, 5:50pm UTC](https://boards.straightdope.com/t/a-maths-questions/194959/1 "2003-08-13T17:50:57Z")
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. ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [August 13, 2003, 6:07pm UTC](https://boards.straightdope.com/t/a-maths-questions/194959/2 "2003-08-13T18:07:31Z")
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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 13, 2003, 6:14pm UTC](https://boards.straightdope.com/t/a-maths-questions/194959/3 "2003-08-13T18:14:40Z")
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. ---
**Author:** ![NE\_Texan](https://avatars.discourse-cdn.com/v4/letter/n/e99b99/32.png) [@NE\_Texan](https://boards.straightdope.com/u/NE_Texan)\ **Post date:** [August 13, 2003, 6:14pm UTC](https://boards.straightdope.com/t/a-maths-questions/194959/4 "2003-08-13T18:14:58Z")
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. ---
**Author:** ![NE\_Texan](https://avatars.discourse-cdn.com/v4/letter/n/e99b99/32.png) [@NE\_Texan](https://boards.straightdope.com/u/NE_Texan)\ **Post date:** [August 13, 2003, 6:17pm UTC](https://boards.straightdope.com/t/a-maths-questions/194959/5 "2003-08-13T18:17:37Z")
I’ll add that any general solution should be a mapping from the interval (1,e) to the interval (e, infinity). ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [August 13, 2003, 6:21pm UTC](https://boards.straightdope.com/t/a-maths-questions/194959/6 "2003-08-13T18:21:01Z")
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. ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [August 13, 2003, 6:34pm UTC](https://boards.straightdope.com/t/a-maths-questions/194959/7 "2003-08-13T18:34:22Z")
Here you go: the [general solution](http://home.earthlink.net/~mrob/pub/math/numbers-3.html#l15_438), algebraically derived.