0\\\\x^{log_{5}x }=(5^{log_{5}x })^{log_{5}x }=5^{log_{5}^{2}x}\\\\5^{log_{5}^{2}x }+x^{log_{5}x }\geq2\sqrt[4]{5} \\\\5^{log_{5}^{2}x}+5^{log_{5}^{2}x}\geq 2\sqrt[4]{5}\\\\2*5^{log_{5}^{2}x}\geq 2\sqrt[4]{5}\\\\5^{log_{5}^{2}x}\geq5^{\frac{1}{4} }\\\\log_{5}^{2} x\geq \frac{1}{4}" alt="1)x>0\\\\x^{log_{5}x }=(5^{log_{5}x })^{log_{5}x }=5^{log_{5}^{2}x}\\\\5^{log_{5}^{2}x }+x^{log_{5}x }\geq2\sqrt[4]{5} \\\\5^{log_{5}^{2}x}+5^{log_{5}^{2}x}\geq 2\sqrt[4]{5}\\\\2*5^{log_{5}^{2}x}\geq 2\sqrt[4]{5}\\\\5^{log_{5}^{2}x}\geq5^{\frac{1}{4} }\\\\log_{5}^{2} x\geq \frac{1}{4}" align="absmiddle" class="latex-formula">
![(log_{5}x-\frac{1}{2})(log_{5}+\frac{1}{2})\geq0\\\\(log_{5}x-log_{5}5^{\frac{1}{2} })(log_{5}x-log_{5}5^{-\frac{1}{2} })\geq0\\\\(5-1)(x-\sqrt{5})(5-1)(x-\frac{1}{\sqrt{5} })\geq0\\\\(x-\sqrt{5})(x-\frac{1}{\sqrt{5} })\geq0\\\\x\in(0;\frac{1}{\sqrt{5} }]\cup[\sqrt{5};+ \infty)](https://tex.z-dn.net/?f=%28log_%7B5%7Dx-%5Cfrac%7B1%7D%7B2%7D%29%28log_%7B5%7D%2B%5Cfrac%7B1%7D%7B2%7D%29%5Cgeq0%5C%5C%5C%5C%28log_%7B5%7Dx-log_%7B5%7D5%5E%7B%5Cfrac%7B1%7D%7B2%7D%20%7D%29%28log_%7B5%7Dx-log_%7B5%7D5%5E%7B-%5Cfrac%7B1%7D%7B2%7D%20%7D%29%5Cgeq0%5C%5C%5C%5C%285-1%29%28x-%5Csqrt%7B5%7D%29%285-1%29%28x-%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D%20%7D%29%5Cgeq0%5C%5C%5C%5C%28x-%5Csqrt%7B5%7D%29%28x-%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D%20%7D%29%5Cgeq0%5C%5C%5C%5Cx%5Cin%280%3B%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D%20%7D%5D%5Ccup%5B%5Csqrt%7B5%7D%3B%2B%20%5Cinfty%29 "(log_{5}x-\frac{1}{2})(log_{5}+\frac{1}{2})\geq0\\\\(log_{5}x-log_{5}5^{\frac{1}{2} })(log_{5}x-log_{5}5^{-\frac{1}{2} })\geq0\\\\(5-1)(x-\sqrt{5})(5-1)(x-\frac{1}{\sqrt{5} })\geq0\\\\(x-\sqrt{5})(x-\frac{1}{\sqrt{5} })\geq0\\\\x\in(0;\frac{1}{\sqrt{5} }]\cup[\sqrt{5};+ \infty)")
3log_{3}x\\\\log_{3}x=m\\\\m^{2}-3m+2>0\\\\(m-1)(m-2)>0\\\\m<1;m>2\\\\log_{3}x<1\\\\x<3\\\\log_{3}x>2\\\\x>9\\\\x\in(0;3)\cup(9;+\infty)" alt="2)log_{3}^{2}x+2>3log_{3}x\\\\log_{3}x=m\\\\m^{2}-3m+2>0\\\\(m-1)(m-2)>0\\\\m<1;m>2\\\\log_{3}x<1\\\\x<3\\\\log_{3}x>2\\\\x>9\\\\x\in(0;3)\cup(9;+\infty)" align="absmiddle" class="latex-formula">
Ответ :
![x\in(0;\frac{1}{\sqrt{5} }]\cup[\sqrt{5};3)\cup(9;+\infty)](https://tex.z-dn.net/?f=x%5Cin%280%3B%5Cfrac%7B1%7D%7B%5Csqrt%7B5%7D%20%7D%5D%5Ccup%5B%5Csqrt%7B5%7D%3B3%29%5Ccup%289%3B%2B%5Cinfty%29 "x\in(0;\frac{1}{\sqrt{5} }]\cup[\sqrt{5};3)\cup(9;+\infty)")