[tex]\displaystyle \sf \sqrt{\ln(x)} +\frac{\sqrt{\ln(x)}}{\ln(x)} = 2023 \\\\ \text{1\º vamos achar as condi\c c\~oes de exist\^encia } : \\\\ \underline{\text{logaritmando maior que 0}}\\\\ x > 0 \\\\ \underline{\text{raiz nos reais \'e maior ou igual a 0}}: \\\\ \sqrt{\ln(x)}\geq 0 \to \ln(x) \geq 0 \to x\geq 1\\\\ \underline{\text{denominador deve ser diferente de 0}}: \\\\ \ln(x) \neq 0\to x\neq e^0 \to x\neq 1[/tex]
[tex]\displaystyle \sf \text{Fazendo as interse\c c\~oes das condi\c c\~oes de exist\^encia} :\\\\ (x > 0)\cap (x\geq 1)\cap (x\neq 1 ) = \large\boxed{\sf x > 1} \\\\ \boxed{\text{Ent\~ao a solu\c c\~ao deve ser maior que 1}}[/tex]
[tex]\displaystyle \sf \text{resolvendo a express\~ao: }\\\\ \sqrt{\ln(x)}+\frac{\sqrt{\ln(x)}}{\ln(x)}=2023 \\\\\\ \text{Mudan\c ca de vari\'avel} : \\\\ \sqrt{\ln(x)} = m \to \ln(x) = m^2\\\\ Assim, \ temos : \\\\ m+\frac{m}{m^2} = 2023 \\\\ \text{multiplique ambos os lados por }m^2 : \\\\ m\cdot m^2+\frac{m\cdot \not m^2}{\not m^2}=2023\cdot m^2\\\\ m^3-2023\cdot m^2+ m = 0 \\\\ m \cdot (m^2-2023m+1) = 0 \\\\ m=0 \to \sqrt{\ln(x)}=0[/tex]
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[tex]\displaystyle \sf \sqrt{\ln(x)} +\frac{\sqrt{\ln(x)}}{\ln(x)} = 2023 \\\\ \text{1\º vamos achar as condi\c c\~oes de exist\^encia } : \\\\ \underline{\text{logaritmando maior que 0}}\\\\ x > 0 \\\\ \underline{\text{raiz nos reais \'e maior ou igual a 0}}: \\\\ \sqrt{\ln(x)}\geq 0 \to \ln(x) \geq 0 \to x\geq 1\\\\ \underline{\text{denominador deve ser diferente de 0}}: \\\\ \ln(x) \neq 0\to x\neq e^0 \to x\neq 1[/tex]
[tex]\displaystyle \sf \text{Fazendo as interse\c c\~oes das condi\c c\~oes de exist\^encia} :\\\\ (x > 0)\cap (x\geq 1)\cap (x\neq 1 ) = \large\boxed{\sf x > 1} \\\\ \boxed{\text{Ent\~ao a solu\c c\~ao deve ser maior que 1}}[/tex]
[tex]\displaystyle \sf \text{resolvendo a express\~ao: }\\\\ \sqrt{\ln(x)}+\frac{\sqrt{\ln(x)}}{\ln(x)}=2023 \\\\\\ \text{Mudan\c ca de vari\'avel} : \\\\ \sqrt{\ln(x)} = m \to \ln(x) = m^2\\\\ Assim, \ temos : \\\\ m+\frac{m}{m^2} = 2023 \\\\ \text{multiplique ambos os lados por }m^2 : \\\\ m\cdot m^2+\frac{m\cdot \not m^2}{\not m^2}=2023\cdot m^2\\\\ m^3-2023\cdot m^2+ m = 0 \\\\ m \cdot (m^2-2023m+1) = 0 \\\\ m=0 \to \sqrt{\ln(x)}=0[/tex]
[tex]\displaystyle \sf \ln(x)=0 \to x= 1\text{ (N\~AO CONV\'EM)}\\ \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\\\ m^2-2023m+ 1 = 0 \\\\ m = \frac{-(-2023)\pm\sqrt{2023^2-4.1.1}}{2.1}\\\\\\ m = \frac{2023\pm\sqrt{2023^2-2^2}}{2}\\\\\ \text{sabemos que }:\\\\ a^2-b^2=(a+b)(a-b) \\\\ 2023^2-2^2=(2023+2)(2023-2)\\\\ 2023^2-2^2=2025.2021 \\ \underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\\\\ 2025 = 45^2[/tex]
[tex]\displaystyle \sf Assim, \\\\ m=\frac{2023\pm\sqrt{45^2\cdot 2021}}{2} \\\\\\ m = \frac{2023\pm 45\sqrt{2021}}{2} \\\\\ \text{para ambos sinais teremos valores positivos. Ent\~ao} : \\\\ m = 2^{-1}.\left(2023+45\sqrt{2021}\right) \\\\\ \sqrt{\ln(x)} = 2^{-1}.\left(2023+45\sqrt{2021}\right) \\\\\ \ln(x) = 4^{-1}.\left(2023+45\sqrt{2021}\right)^2[/tex]
[tex]\displaystyle \sf \boxed{\sf \ x = \left(e\right)^{4^{-1}.\left(2023+45\sqrt{2021}\right)^2 }}\ ,\ x > 1 \checkmark \\\\\ e \\\\\ \boxed{\sf \ x = \left(e\right)^{{4^{-1}.\left(2023-45\sqrt{2021}\right)^2 }}}\ ,\ x > 1 \checkmark[/tex]