Equazioni logaritmiche riassuntive.

LN(x) = t  , con x>0

t^2 - t - 12 = 0

(t + 3)·(t - 4) = 0

t = 4 ∨ t = -3

LN(x) = 4---> x = e^4

LN(x) = -3---> x = e^(-3)

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$\small \left(ln(x)\right)^2-ln(x)-12=0$

$\small \left(ln_e(x)\right)^2-ln_e(x)-12=0$

$\small ln_e(x) = t$

$\small t^2-t-12=0$

$\small a= 1; b= -1 c= -12$

$\small \Delta= b^2-4ac = (-1)^2-(4·1·-12) = 1-(-48) = 1+48=49$

$\small t_{1,2}= \dfrac{-b\pm\sqrt{\Delta}}{2a} = \dfrac{-(-1)\pm\sqrt{49}}{2·1}=\dfrac{1\pm7}{2}$

$\small t_1= \dfrac{1-7}{2} = \dfrac{-6}{2} = -3$

$\small t_2= \dfrac{1+7}{2} = \dfrac{8}{2} = 4$

quindi:

$\small t_1= ln_e(x) =  -3 \quad\Longrightarrow x= e^{-3};$

$\small t_2= ln_e(x) =  4 \quad\Longrightarrow x= e^4.$