Resposta:

(2^x) * 3^(x²)=6

log (2^x) * 3^(x²)= log 6

log (2^x) + log 3^(x²)= log 6

x*log 2 + x²* log 3= log 6  

x'=[-log(2)+√(log²(2) -4*log(3)*(-log(6))]/2log(3)

x'=[-log(2)+√(log²(2) +4*log(3)*(log(3)+log(2)))]/2log(3)

x'=[-log(2)+√(log²(2) +4*log²(3)+4*log(3)*log(2))]/2log(3)

x'=[-log(2)+√(log(2) +2*log(3))²)]/2log(3)

x'=[-log(2)+log(2) +2*log(3))]/2log(3)

x'=[2log(3))]/2log(3)

x'=1

x''=[-log(2)-√(log²(2) -4*log(3)*(-log(6))]/2log(3)

x''=[-log(2)-√(log²(2) +4*log(3)*(log(3)+log(2)))]/2log(3)

x''=[-log(2)-√(log²(2) +4*log²(3)+4*log(3)*log(2))]/2log(3)

x''=[-log(2)-√(log(2) +2*log(3))²)]/2log(3)

x''=[-log(2)-log(2) -2*log(3))]/2log(3)

x''=[-log(2)) - log(3))]/log(3)

x''=-log(6)/log(3) = -log₃ 6

[tex]\Large\begin{array}{c}2^x\cdot 3^{x^2}=6\end{array} \\ \Large\begin{array}{c}2^x \cdot2 {}^{ log_{2}(3) {x}^{2} } =6\end{array} \\ \Large\begin{array}{c}2^{x \: + \: log_{2}(3) {x}^{2} }=6\end{array} \\ \Large\begin{array}{c} In \left(_2x + log_{2}(3) {x}^{2} \right) = In(6)\end{array} \\ [/tex]

[tex]\Large\begin{array}{c}(x + log_{2}(3) {x}^{2})In(2) = In(6) \end{array} \\ \Large\begin{array}{c}In(2) x + In(3) {x}^{2} = In(6) \end{array} \\ \color{green}\Large\begin{array}{c}In(3) x {}^{2} + In(2) {x} - In(6) = 0 \end{array}[/tex]

[tex]\Large\begin{array}{c}a = In(3) \\ b = In(2) \\ c = - In(6) \end{array}[/tex]

Regras normais de equação do segundo grau

[tex] \Delta = {b}^{2} - 4 \times a \times c \\ \Large\begin{array}{c} \Delta = In(2)² - 4 \times In(3) \times ( - In(6)) \end{array} \\ \Large\begin{array}{c} \color{green}\Delta = In(2)² + 4 \times In(3) \times In(6) \end{array} \\ ou \\ \Delta ≈8.35425[/tex]

Temos 2 raízes Reais

[tex]\Large\begin{array}{c} \frac{ - In(2)± \sqrt{ In(2)² + 4 \times In(3) \times In(6)} }{2In(3)}\end{array} \begin{gathered}\begin{cases} { x_2 = \Large\begin{array}{c} \frac{ - In(2) + \sqrt{ In(2)² + 4 \times In(3) \times In(6)} }{2In(3)}\end{array}}\\ \\ { x_1 = \Large\begin{array}{c} \frac{ - In(2) - \sqrt{ In(2)² + 4 \times In(3) \times In(6)} }{2In(3)}\end{array} } \end{cases}\end{gathered} \\ ou \\ x_2 = 1 \\ x_1 = - 1.63093[/tex]

[tex]{\huge\boxed { {\bf{E}}}\boxed { \red {\bf{a}}} \boxed { \blue {\bf{s}}} \boxed { \gray{\bf{y}}} \boxed { \red {\bf{}}} \boxed { \orange {\bf{M}}} \boxed {\bf{a}}}{\huge\boxed { {\bf{t}}}\boxed { \red {\bf{h}}}} \\ \boxed{ \displaystyle\int_ \empty ^ \mathbb{C}     \frac{ - b \: ± \:  \sqrt{ {b}^{2} - 4 \times a \times c } }{2 \times a} d{ t } \boxed{ \boxed{ \mathbb{\displaystyle\Re}\sf{ \gamma  \alpha }\tt{ \pi}\bf{ \nabla}}}} \\ {\boxed{ \color{blue} \boxed{ 26 |12|22 }}}{\boxed{ \color{blue} \boxed{Espero \: ter \: ajudado \: ☆}}}[/tex]