LOG(3,√x)^2 = 2·LOG(3,x^(1/3))
cambio base:
LN(x)^2/(4·LN(3)^2) = 2·LN(x)/(3·LN(3))
(*12·LN(3)^2)
3·LN(x)^2 = 8·LN(3)·LN(x)
LN(x) = t
3·t^2 = 8·LN(3)·t
3·t^2 - 8·LN(3)·t = 0
t·(3·t - 8·LN(3)) = 0
t = 8·LN(3)/3 ∨ t = 0
LN(x) = 8·LN(3)/3: x = 9·3^(2/3)
LN(x) = 0 : x = 1
$ log_3^2 \sqrt{x} = 2log_3 \sqrt[3] x $
$ \frac{1}{4} (log_3 x)^2 - \frac{2}{3}log_3 x = 0 $
Poniamo $t = log_3 x $
$ \frac{1}{4} t^2 - \frac{2}{3} t = 0 $
$ 3t^2 - 8t = 0 $
$ t(3t-8) = 0 $
Due soluzioni
$\; ⇒ \; log_3 x = 2+\frac{2}{3} \; ⇒ \; x = 9 \sqrt[3]{9} $