[tex]\displaystyle \sf \sqrt[3]{\sf \log(x)} = \log(\sqrt[3]{\sf x}) \\\\ \sqrt[3]{\sf \log(x)} = \log(x^{\frac{1}{3}} ) \\\\ \sqrt[3]{\sf \log(x)} = \frac{1}{3}\cdot \log(x) \\\\ \left(\sqrt[3]{\sf \log(x)} \right)^{3} =\left( \frac{\log(x)}{3}\right)^{3} \\\\ \log(x) = \frac{\log^3(x) }{27} \\\\ 27\cdot \log(x) = \log^3(x) \\\\\ \log^3(x) -27\log(x) =0\\\\\ \log(x)\cdot \left[\log^2(x) -27\right] = 0 \\\\\ \log(x) =0 \to x=10^0 \to \boxed{\sf x = 1 } \\\\[/tex]

[tex]\displaystyle \sf \log^2(x) -27 = 0 \to \log^2(x) = 27 \to \log(x) = \pm\sqrt{27} \\\\\ \log(x) = \pm 3\sqrt{3} \to \log(x) = 3\sqrt{3} \to \boxed{\sf x = 10^{3\sqrt{3}} } \\\\\ \log(x) =-3\sqrt{3} \to \log(x) =-3\sqrt{3} \to \boxed{\sf x= 10^{-3\sqrt{3}} }[/tex]

Portanto o conjunto solução é :

[tex]\displaystyle \sf \large\boxed{\sf \ x = 1 \ \ ;\ \ x=10^{3\sqrt{3}} \ \ ;\ \ x=\frac{1}{10^{3\sqrt{3}}} \ }\checkmark[/tex]

Verified answer

Explicação passo-a-passo:

[tex] \sqrt[3]{ log(x) } = log( \sqrt[3]{x} ) \\ \sqrt[3]{ log(x) } = \frac{1}{3} \times log(x) \\ ( \sqrt[3]{ log(x) } ) {}^{3} = ( \frac{1}{3} \times log(x) ) {}^{3} \\ \\ log(x) = ln(x) = y \\ \\ \sqrt[3]{y} {}^{3} = ( \frac{y}{3} ) {}^{3} \\ y = \frac{y {}^{3} }{27} \\ 27y = y {}^{3} \\ y {}^{3} - 27y = 0 \\ y(y {}^{2} - 27) = 0 \\ \\ y = 0 \\ \\ y {}^{2} - 27 = 0 \\ y {}^{2} = 27 \\ \sqrt{y {}^{2} } = \sqrt{27} \\ |y| = \sqrt{9 \times 3} \\ y = - 3 \sqrt{3} \\ y = 3 \sqrt{3} \\ \\ ln(x) = y \\ ln(x) = 0 \\ x = e {}^{0} \\ x = 1 \\ \\ ln(x) = y \\ ln(x) = - 3 \sqrt{3} \\ x = e {}^{ - 3 \sqrt{3} } \\ x = \frac{1}{e {}^{3 \sqrt{3} } } \\ \\ ln(x) = y \\ ln(x) = 3 \sqrt{3} \\ x = e {}^{3 \sqrt{3} } \\ \\ x = 1 \\ x = \frac{1}{e {}^{3 \sqrt{3} } } \\ x = e {}^{3 \sqrt{3} } [/tex]