How do you solve for x algebraically?

Question

$$x^{\log2 / \log3} = x^{1/2} + 1$$

Using desmos, I figured out that the answer is 9. But is there a way to solve algebraically??

Answer 1

$$x^{\frac{\log2} {\log3}} = x^{\frac12} + 1$$ Let $y=x^{\frac12}$, $$y^{\frac{2\log2} {\log3}} = y + 1$$ $$\frac{2\log2} {\log3}\log y =\log \left( y + 1\right)$$ $${\log y}\cdot\log 4 =\log 3 \cdot\log \left( y + 1\right)$$

Compare two sides, $$\therefore y= 3,x=y^2=9.$$

Answer 2

$x=9$ is the only solution because $x=9$ is the only intersection between $x$ and $1-2x^k+x^{2k}$, where $k=\frac{log2}{log3}$.