Question

log3 (x^2 − x − 5) − log3 x = 1
alguem sabe como resolver?
a resp é 5

Answer 1

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$\sf\ell og_3(x^2-x-5)-\ell og_3x=1\\\sf\ell og_3\left(\dfrac{x^2-x-6}{x}\right)=1\\\sf\dfrac{x^2-x-5}{x}=3^1\\\sf\dfrac{x^2-x-5}{x}=3\\\sf x^2-x-5=3x\\\sf x^2-x-3x-5=0\\\sf x^2-4x-5=0$

$\sf\Delta=b^2-4ac\\\sf\Delta=(-4)^2-4\cdot1\cdot(-5)\\\sf\Delta=16+20\\\sf\Delta=36\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\sf x=\dfrac{-(-4)\pm\sqrt{36}}{2\cdot1}\\\sf x=\dfrac{4\pm6}{2}\begin{cases}\sf x_1=\dfrac{4+6}{2}=\dfrac{10}{2}=5\\\sf x_2=\dfrac{4-6}{2}=-\dfrac{2}{2}=-1\end{cases}\\\sf como~x>0~temos:\\\huge\boxed{\boxed{\boxed{\boxed{\sf x=5\checkmark}}}}$