How to see that $1$ is a solution of $x^{x^2−3x} = x^2$

Question

I tried to solve the problem below to get all the positive solutions: $$x^{x^2−3x} = x^2$$

By using $\ln$ on both sides, I get that one solution is $\displaystyle\frac{3 + \sqrt{17}}{2}$. But $1$ is also a solution that you can guess. How can I see it while solving the equation?

EDIT: I had a typo that showed i in front of $\displaystyle \frac{3 + \sqrt{17}}{2}$.

Is it more obvious if you rewrite the equation as $x^{x^2 - 3x - 2} = 1$? — Willie Wong, May 10 '19 at 05:26
$\mathrm i\bigl(3+\sqrt{17}\bigr)/2$ can’t be a “positive solution” because it isn’t real. It lies entirely on the imaginary axis, and $\Im\left{\mathrm i\bigl(3+\sqrt{17}\bigr)/2\right}$ is positive, if that’s what you mean. — gen-ℤ ready to perish, May 10 '19 at 07:54

score 5 · Accepted Answer · answered May 10 '19 at 05:25

5

$$x^{x^2−3x} = x^2$$

Using $\ln$ on both sides, $$\implies (x^2-3x)\cdot\ln x = 2 \ln x$$

Here, you can divide by $\ln x$ if $\ln x \ne 0$, i.e., if $x \ne 1$. You have assumed that case while equating the powers.

answered May 10 '19 at 05:25

user1952500

2,121

David G. Stork · Answer 2 · 2019-05-10T05:54:26.137

2

Take the $\ln$ of both sides.

You have $\ln x$ on each side of an equality and this immediately gives $x = 1$ as a solution. Factor out that $\ln x$ (for $x \neq 1$ solutions) and get a simple quadratic with imaginary solutions:

$$x = \frac{i}{2} \left(3 \pm \sqrt{17}\right)$$

I'd love to hear the justification of the downvote of my solution. Please post it as a comment, whoever you are.

edited May 10 '19 at 05:54

answered May 10 '19 at 05:39

David G. Stork

29,774

I'm m not the one who downvoted your answer, but I'll upvote it because it makes sense. – May 10 '19 at 05:55
1

Welcome to the club of victims of mysterious (and for sure anonymous) downvoters ! $\to +1$. – Claude Leibovici May 10 '19 at 06:15
I voted up it as well to compensate. Thanks for replying :) – Dovendyr May 10 '19 at 13:16

How to see that $1$ is a solution of $x^{x^2−3x} = x^2$

2 Answers2