Question

[ log (x) ] ^2 - log (x^2) - 3 =0 ?

Answer 1

$(logx)^{2}-logx^{2}-3 = 0 \\\\ (logx)^{2}-2 \cdot logx-3 = 0 \\\\ \boxed{logx = t} \\\\ t^{2}-2t-3 = 0 \\\\ \Delta = b^{2}-4ac \\\\ \Delta = (-2)^{2}-4 \cdot (1) \cdot (-3) \\\\ \Delta = 4+12 \\\\ \Delta = 16$

$t = \frac{-b \pm \sqrt{\Delta}}{2a} \\\\ t = \frac{-(-2) \pm \sqrt{16}}{2 \cdot 1} \\\\ t = \frac{2 \pm 4}{2} \\\\\\ \Rightarrow t' = \frac{2+4}{2} = \frac{6}{2} = 3 \\\\ \Rightarrow t'' = \frac{2-4}{2} = -\frac{2}{2} = -1$

Voltando:

$\boxed{logx = t} \\\\ \rightarrow logx = 3 \ \ \Rightarrow x = 10^{3} \ \ \Rightarrow \boxed{\boxed{x = 1.000}} \\\\ \Rightarrow logx = -1 \ \ \Rightarrow x = 10^{-1} \ \ \Rightarrow \boxed{\boxed{x = 0,1}}$

A solução é x valendo 1.000 ou 0,1.