Question

27 vezes x^logx na base 3=x^4

Answer 1

$\mathrm{27x^{\log_3{x}}=x^4\ \to\ x^{\log_3{x}}=\dfrac{x^4}{27}\ \to\ \log_3{(x^{\log_3{x}})}=\log_3{\bigg(\dfrac{x^4}{27}\bigg)}}\\\\ \mathrm{(\log_3{x})(\log_3{x})=\log_3{x^4}-\log_3{27}\ \to\ (\log_3{x})^2=4\log_3{x}-3}$

$\mathrm{\Rightarrow Fazendo\ \boxed{\mathrm{\log_3{x}=u}}\ \to\ u^2=4u-3\ \to\ u^2-4u+3=0}\\\\ \mathrm{u=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}=\dfrac{-(-4)\pm\sqrt{(-4)^2-4.1.3}}{2.1}=}\\\\ \mathrm{=\dfrac{4\pm\sqrt{16-12}}{2}=\dfrac{4\pm\sqrt{4}}{2}=\dfrac{4\pm2}{2}=2\pm1=\boxed{1}\ ou\ \boxed{3}}$

$\mathrm{\Rightarrow Para\ \boxed{\mathrm{u=1}}\ \to\ \log_3{x}=1\ \to\ \boxed{\boxed{\mathbf{x=3}}}}\\\\ \mathrm{\Rightarrow Para\ \boxed{\mathrm{u=3}}\ \to\ \log_3{x}=3\ \to\ \boxed{\boxed{\mathbf{x=27}}}}\\\\ \textbf{Solu\c{c}\~ao:}\ \mathrm{x=\{3;27\}}$