Question

Resolva a equação exponencial.

Answer 1

$\Large\boxed{\begin{array}{l}\sf 9\cdot x^{\log_3x}=x^3\\\sf fac_{\!\!,}a~log_3x=y\implies x=3^y.\\\sf 9\cdot(3^y)^y=(3^y)^3\\\sf 9\cdot3^{y^2}=3^{3y\\\sf 3^2\cdot3^{y^2}}=3^{3y}\\\sf 3^{y^2+2}=3^{3y}\\\sf y^2+2=3y\end{array}}$

$\large\boxed{\begin{array}{l}\sf y^2-3y+2=0\\\sf\Delta=b^2-4ac\\\sf\Delta=(-3)^2-4\cdot1\cdot2\\\sf\Delta=9-8\\\sf\Delta=1\\\sf y=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf y=\dfrac{-(-3)\pm\sqrt{1}}{2\cdot1}\\\\\sf y=\dfrac{3\pm1}{2}\begin{cases}\sf y_1=\dfrac{3+1}{2}=\dfrac{4}{2}=2\\\\\sf y_2=\dfrac{3-1}{2}=\dfrac{2}{2}=1\end{cases}\end{array}}$

$\Large\boxed{\begin{array}{l}\sf para~y=2\\\sf x=3^y\\\sf x=3^2\\\sf x=9\\\sf para~y=1\\\sf x=3^1\\\sf x=3\\\sf S=\{3,9\}\end{array}}$