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July 2022 1 17 Report

Решите неравенства:
а). [tex] log ^{2} _{2} x^{2} -15 log_{2}x-4 \leq 0 [/tex]
б). [tex] log ^{2} _{ \frac{1}{3} } x^{2} -7 log_{ \frac{1}{3} }x+3 \leq 0 [/tex]
в). [tex] log ^{2} _{ 3} x^{2} +13 log_{ 3 }x+3 < 0 [/tex]
г). [tex] log ^{2} _{ \frac{1}{5} } x^{2} -31 log_{ \frac{1}{5} }x-8 < 0 [/tex]

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arsenlevadniy

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$\log ^{2}_{2}x^{2}-15\log_{2}x-4 \leq 0, \\ 2\log ^{2}_{2}x-15\log_{2}x-4 \leq 0,, \\ \log_{2}x=t, \\ 2t^2-15t-4 \leq 0, \\ 2t^2-15t-4=0, \\ D=257>0, \\ a=2>0, \\ t_1=\frac{15-\sqrt{257}}{4}, t_2=\frac{15+\sqrt{257}}{4}, \\ \frac{15-\sqrt{257}}{4} \leq t \leq \frac{15+\sqrt{257}}{4}, \\ \frac{15-\sqrt{257}}{4} \leq \log_{2}x \leq \frac{15+\sqrt{257}}{4}, \\ 2^{\frac{15-\sqrt{257}}{4}} \leq x \leq 2^{\frac{15+\sqrt{257}}{4}}.$

$\log ^{2}_{\frac{1}{3}}x^{2}-7\log_{\frac{1}{3}}x+3 \leq 0, \\ 2\log ^{2}_{\frac{1}{3}}x-7\log_{\frac{1}{3}}x+3 \leq 0, \\ \log_{\frac{1}{3}}x=t, \\ 2t^2-7t+3 \leq 0, \\ 2t^2-7t+3 \leq 0, \\ D=25=5^2>0, \\ a=2>0, \\ t_1=\frac{1}{2}, t_2=3, \\ \frac{1}{2} \leq t \leq 3, \\ \frac{1}{2} \leq \log_{\frac{1}{3}}x \leq 3, \\ (\frac{1}{3})^{\frac{1}{2}} \geq x \geq (\frac{1}{3})^{3}, \\ \frac{1}{27} \leq x \leq \frac{\sqrt{3}}{3}.$

$\log ^{2}_{3}x^{2}+13\log_{3}x+3 < 0, \\ 2\log ^{2}_{3}x+13\log_{3}x+3 < 0, \\ \log_{3}x = t, \\ 2t^2+13t+3 < 0, \\ D=145>0, \\ a=2>0, \\ t_1=\frac{-13-\sqrt{145}}{4}, t_2=\frac{-13+\sqrt{145}}{4}, \\ \frac{-13-\sqrt{145}}{4} < t < \frac{-13+\sqrt{145}}{4}, \\ \frac{-13-\sqrt{145}}{4} < \log_{3}x < \frac{-13+\sqrt{145}}{4}, \\ 3^{\frac{-13-\sqrt{145}}{4}} < x < 3^{\frac{-13+\sqrt{145}}{4}}.$

$\log^{2}_{\frac{1}{5}}x^{2}-31\log_{\frac{1}{5}}x-8 < 0, \\ 2\log^{2}_{\frac{1}{5}}x-31\log_{\frac{1}{5}}x-8 < 0, \\ \log_{\frac{1}{5}}x=t, \\ 2t^2-31t-8 < 0, \\ D=1025=25\cdot41>0, \\ a=2>0, \\ t_1=\frac{31-5\sqrt{41}}{4}, t_2=\frac{31+5\sqrt{41}}{4}, \\ \frac{31-5\sqrt{41}}{4} < t < \frac{31+5\sqrt{41}}{4}, \\ \frac{31-5\sqrt{41}}{4} < \log_{\frac{1}{5}}x < \frac{31+5\sqrt{41}}{4}, \\ (\frac{1}{5})^{\frac{31-5\sqrt{41}}{4}} > x > (\frac{1}{5})^{\frac{31+5\sqrt{41}}{4}}.$

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