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Прологарифмируем обе части по основанию 10 Сделаем замену lg²x = m ≥ 0 , ОДЗ: x > 0 9m² - 2m - 7 = 0 D = (-2)² - 4 * 9 * (- 7) = 4 + 252 = 256 = 16² m₂ < 0 - не подходит lg²x = 1 или lgx = 1 ⇒ x = 10 или lgx = - 1 ⇒ x = 0,1
![lgx ^{3lg ^{3}x- \frac{2}{3}lgx }=lg100 \sqrt[3]{10}\\\\(3lg ^{3}x- \frac{2}{3}lgx)lgx=lg10 ^{2}*10 ^{ \frac{1}{3} }\\\\3lg ^{4}x- \frac{2}{3}lg ^{2} x=lg10 ^{ \frac{7}{3} }\\\\3lg ^{4}x- \frac{2}{3}lg ^{2}x - \frac{7}{3}=0\\\\9lg ^{4} x-2lg ^{2}x-7=0](https://tex.z-dn.net/?f=lgx+%5E%7B3lg+%5E%7B3%7Dx-+%5Cfrac%7B2%7D%7B3%7Dlgx++%7D%3Dlg100+%5Csqrt%5B3%5D%7B10%7D%5C%5C%5C%5C%283lg+%5E%7B3%7Dx-+%5Cfrac%7B2%7D%7B3%7Dlgx%29lgx%3Dlg10+%5E%7B2%7D%2A10+%5E%7B+%5Cfrac%7B1%7D%7B3%7D+%7D%5C%5C%5C%5C3lg+%5E%7B4%7Dx-+%5Cfrac%7B2%7D%7B3%7Dlg+%5E%7B2%7D+x%3Dlg10+%5E%7B+%5Cfrac%7B7%7D%7B3%7D+%7D%5C%5C%5C%5C3lg+%5E%7B4%7Dx-+%5Cfrac%7B2%7D%7B3%7Dlg+%5E%7B2%7Dx+-+%5Cfrac%7B7%7D%7B3%7D%3D0%5C%5C%5C%5C9lg+%5E%7B4%7D+x-2lg+%5E%7B2%7Dx-7%3D0+++++++++++++ "lgx ^{3lg ^{3}x- \frac{2}{3}lgx }=lg100 \sqrt[3]{10}\\\\(3lg ^{3}x- \frac{2}{3}lgx)lgx=lg10 ^{2}*10 ^{ \frac{1}{3} }\\\\3lg ^{4}x- \frac{2}{3}lg ^{2} x=lg10 ^{ \frac{7}{3} }\\\\3lg ^{4}x- \frac{2}{3}lg ^{2}x - \frac{7}{3}=0\\\\9lg ^{4} x-2lg ^{2}x-7=0")
