0, \ x \ne 1\\\\
3\lg^2 x = 1 + 2\lg x\\\\
3\lg^2 x - 2\lg x - 1 = 0, \ t = \lg x" alt="x^{3\lg x} = 10x^2\\\\
10 = x^{\log_{x} 10}\\\\
x^{3\lg x} = x^{\log_{x} 10}x^2\\\\
x^{3\lg x} = x^{\log_{x} 10 + 2}\\\\
3\lg x = \log_{x}10 + 2 \ | \ * \lg x, \ x > 0, \ x \ne 1\\\\
3\lg^2 x = 1 + 2\lg x\\\\
3\lg^2 x - 2\lg x - 1 = 0, \ t = \lg x" align="absmiddle" class="latex-formula">
![3t^2 - 2t - 1 = 0\\\\
3t^2 - 3t + t - 1 = 0\\\\
3t(t - 1) + 1(t - 1) = 0\\\\
(3t + 1)(t - 1) = 0\\\\ 1) \
3t + 1 = 0, \ 3t = -1, \ t = -\frac{1}{3}\\\\ \lg x = -\frac{1}{3}, \ \boxed{x = \frac{1}{\sqrt[3]{10}} }\\\\
2) \ t - 1 = 0, \ t = 1\\\\
\lg x = 1, \ \boxed{x = 10}](https://tex.z-dn.net/?f=3t%5E2+-+2t+-+1+%3D+0%5C%5C%5C%5C%0A3t%5E2+-+3t+%2B+t+-+1+%3D+0%5C%5C%5C%5C%0A3t%28t+-+1%29+%2B+1%28t+-+1%29+%3D+0%5C%5C%5C%5C%0A%283t+%2B+1%29%28t+-+1%29+%3D+0%5C%5C%5C%5C+1%29+%5C%0A3t+%2B+1+%3D+0%2C+%5C+3t+%3D+-1%2C+%5C+t+%3D+-%5Cfrac%7B1%7D%7B3%7D%5C%5C%5C%5C+%5Clg+x+%3D+-%5Cfrac%7B1%7D%7B3%7D%2C+%5C+%5Cboxed%7Bx+%3D+%5Cfrac%7B1%7D%7B%5Csqrt%5B3%5D%7B10%7D%7D+%7D%5C%5C%5C%5C%0A2%29+%5C+t+-+1+%3D+0%2C+%5C+t+%3D+1%5C%5C%5C%5C%0A%5Clg+x+%3D+1%2C+%5C+%5Cboxed%7Bx+%3D+10%7D "3t^2 - 2t - 1 = 0\\\\
3t^2 - 3t + t - 1 = 0\\\\
3t(t - 1) + 1(t - 1) = 0\\\\
(3t + 1)(t - 1) = 0\\\\ 1) \
3t + 1 = 0, \ 3t = -1, \ t = -\frac{1}{3}\\\\ \lg x = -\frac{1}{3}, \ \boxed{x = \frac{1}{\sqrt[3]{10}} }\\\\
2) \ t - 1 = 0, \ t = 1\\\\
\lg x = 1, \ \boxed{x = 10}")