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1-2-log-9-x-log-3-5x-gt-log-1-3-x-1-

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Question Number 4592 by love math last updated on 09/Feb/16
(1/2)+log_9 x−log_3 5x>log_(1/3) (x+1)
$$\frac{\mathrm{1}}{\mathrm{2}}+{log}_{\mathrm{9}} {x}−{log}_{\mathrm{3}} \mathrm{5}{x}>{log}_{\frac{\mathrm{1}}{\mathrm{3}}} \left({x}+\mathrm{1}\right) \\ $$
Answered by Rasheed Soomro last updated on 10/Feb/16
(1/2)+log_9 x−log_3 5x>log_(1/3) (x+1) (1/2)+((log_3 x)/(log_3 9))−((log_3 5x)/(log_3 3))>((log_3 (x+1))/(log_3 (1/3))) (1/2)+((log_3 x)/2)−((log_3 5x)/1)>((log_3 (x+1))/(−1)) 1+log_3 x−2log_3 5x>−2log_3 (x+1) log_3 3+log_3 x_(−) −log_3 (5x)^2 >log_3 (x+1)^(−2) log_3 3x_(−) −log_3 25x^2 >log_3 (x+1)^(−2) log_3 (((3x)/(25x^2 )))>log_3 ((1/((x+1)^2 ))) (3/(25x))>(1/((x+1)^2 )) (3/(25x))−(1/((x+1)^2 ))>0 ((3(x+1)^2 −25x)/(25x(1+x)^2 ))>0 ((3x^2 +6x+3−25x)/(25x(1+x)^2 ))>0 ((3x^2 −19x+3)/(25x(1+x)^2 ))>0 (N/D)>0⇒ N ,D >0 ∣ N ,D<0 When N,D>0 3x^2 −19x+3>0 ∧ 25x(1+x)^2 >0 25x(1+x)^2 >0⇒x>0 When N,D<0 3x^2 −19x+3<0 ∧ 25x(1+x)^2 <0 Continue
$$\frac{\mathrm{1}}{\mathrm{2}}+{log}_{\mathrm{9}} {x}−{log}_{\mathrm{3}} \mathrm{5}{x}>{log}_{\frac{\mathrm{1}}{\mathrm{3}}} \left({x}+\mathrm{1}\right) \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}+\frac{{log}_{\mathrm{3}} \:{x}}{{log}_{\mathrm{3}} \:\mathrm{9}}−\frac{{log}_{\mathrm{3}} \:\mathrm{5}{x}}{{log}_{\mathrm{3}} \:\mathrm{3}}>\frac{{log}_{\mathrm{3}} \:\left({x}+\mathrm{1}\right)}{{log}_{\mathrm{3}} \:\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}+\frac{{log}_{\mathrm{3}} \:{x}}{\mathrm{2}}−\frac{{log}_{\mathrm{3}} \:\mathrm{5}{x}}{\mathrm{1}}>\frac{{log}_{\mathrm{3}} \:\left({x}+\mathrm{1}\right)}{−\mathrm{1}} \\ $$$$\mathrm{1}+{log}_{\mathrm{3}} {x}−\mathrm{2}{log}_{\mathrm{3}} \mathrm{5}{x}>−\mathrm{2}{log}_{\mathrm{3}} \left({x}+\mathrm{1}\right) \\ $$$$\underset{−} {{log}_{\mathrm{3}} \mathrm{3}+{log}_{\mathrm{3}} {x}}−{log}_{\mathrm{3}} \left(\mathrm{5}{x}\right)^{\mathrm{2}} >{log}_{\mathrm{3}} \left({x}+\mathrm{1}\right)^{−\mathrm{2}} \\ $$$$\underset{−} {{log}_{\mathrm{3}} \mathrm{3}{x}}−{log}_{\mathrm{3}} \mathrm{25}{x}^{\mathrm{2}} >{log}_{\mathrm{3}} \left({x}+\mathrm{1}\right)^{−\mathrm{2}} \\ $$$${log}_{\mathrm{3}} \left(\frac{\mathrm{3}{x}}{\mathrm{25}{x}^{\mathrm{2}} }\right)>{log}_{\mathrm{3}} \left(\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }\right) \\ $$$$\frac{\mathrm{3}}{\mathrm{25}{x}}>\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\frac{\mathrm{3}}{\mathrm{25}{x}}−\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }>\mathrm{0} \\ $$$$\frac{\mathrm{3}\left({x}+\mathrm{1}\right)^{\mathrm{2}} −\mathrm{25}{x}}{\mathrm{25}{x}\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }>\mathrm{0} \\ $$$$\frac{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{3}−\mathrm{25}{x}}{\mathrm{25}{x}\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }>\mathrm{0} \\ $$$$\frac{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{19}{x}+\mathrm{3}}{\mathrm{25}{x}\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }>\mathrm{0} \\ $$$$\frac{{N}}{{D}}>\mathrm{0}\Rightarrow\:{N}\:,{D}\:>\mathrm{0}\:\mid\:{N}\:,{D}<\mathrm{0}\: \\ $$$${When}\:{N},{D}>\mathrm{0} \\ $$$$\mathrm{3}{x}^{\mathrm{2}} −\mathrm{19}{x}+\mathrm{3}>\mathrm{0}\:\wedge\:\mathrm{25}{x}\left(\mathrm{1}+{x}\right)^{\mathrm{2}} >\mathrm{0} \\ $$$$\mathrm{25}{x}\left(\mathrm{1}+{x}\right)^{\mathrm{2}} >\mathrm{0}\Rightarrow{x}>\mathrm{0} \\ $$$$ \\ $$$$ \\ $$$${When}\:{N},{D}<\mathrm{0} \\ $$$$\:\mathrm{3}{x}^{\mathrm{2}} −\mathrm{19}{x}+\mathrm{3}<\mathrm{0}\:\:\wedge\:\mathrm{25}{x}\left(\mathrm{1}+{x}\right)^{\mathrm{2}} <\mathrm{0} \\ $$$$ \\ $$$${Continue} \\ $$

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