# If $x^y = xy$, what is $y$ in terms of $x$?

I know that $$x=y^{\frac{1}{y-1}}$$ but I cannot solve for $$x$$. Can somebody please help me? It should be possible because it is a very simple equation, can somebody please solve it.

• Have you come across logs? – Olly Reynolds Apr 30 at 12:53
• How are $x$ and $y$ defined? Also, don't assume that because some expression looks simple it should be fit for understanding using only elementary means. Not always the case. Also, you do not define what it means to express a variable in terms of others. What operations are permitted -- beyond radicals, logs, series expansions, etc.? – Allawonder Apr 30 at 13:15
• @Jamesodare How do you propose to use logs to express $y$ in terms of $x$ here? – Allawonder Apr 30 at 13:15
• @Jamesodare please tell me how you can use logs to solve for y in terms of x, I do not understand how you are supposed to do that. – user604253 Apr 30 at 14:12

## 1 Answer

As I say in the comments, many things are undefined, but if $$x$$ never disappears, then we can write $$y=x^{y-1},$$ which is equivalent to $$y-1=\log_x y.$$ We then see that $$x,y$$ should satisfy the conditions $$1\ne x>0,y>0$$ for uniquely defined values of $$y$$ regarded as a real-valued function of the real variable $$x.$$

Also, it appears that there's no elementary way to express $$y$$ in terms of $$x.$$

• please tell me if there is any way to solve for y even if it is not an elementary way. – user604253 Apr 30 at 14:10
• @datboi Change base of $text{RHS}$ to $e,$ expand as a power series (this limits the domain), and invert this series. Good luck. – Allawonder Apr 30 at 17:22
• please explain a bit more – user604253 Apr 30 at 19:41
• @datboi Do you know calculus? Are you familiar with power series? If not, no matter how I explain it will be wasted. Try to understand these topics first. Why do you need to express $y$ in terms of $x$ so much -- one can't always do some things in mathematics, just as in life. Even if you use the power series as I suggested, it will only be valid in an interval, not the whole domain; and to use it elsewhere you'll always have to make appropriate changes. What do you want to do so badly with an expression $y=f(x)$? There may be better methods. – Allawonder May 1 at 6:40