no. and nature of roots of $x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$

Question

The given equation is

$$x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$$

I took $\log_{2}{x}$ = $t$

and then rewrote the given equation as

$$x^{3t^2 + 4t - 5} = \sqrt{2}$$

But I don't know what to do after this. How will I find the nature and no. of roots?

Answer 1

$$x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$$ $$\log_2 {x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}}} = \log_2 {\sqrt{2}}$$ $$\log_{2}{x} ({\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}})=\frac12$$ $t=\log_2{x}$ $$3t^3+4t^2-5t-2=0$$ $t_1=1.$ Can you finish?

Answer 2

I'd try to take $\log_2$ of the whole expression and the solve with respect to $t=\log_2 x$.