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log-x-x-3-7-lt-log-x-3-7-




Question Number 85694 by john santu last updated on 24/Mar/20
log_(((x/(x−3)))) (7) < log_(((x/3)))  (7)
$$\mathrm{log}_{\left(\frac{{x}}{{x}−\mathrm{3}}\right)} \left(\mathrm{7}\right)\:<\:\mathrm{log}_{\left(\frac{{x}}{\mathrm{3}}\right)} \:\left(\mathrm{7}\right)\: \\ $$
Commented by jagoll last updated on 24/Mar/20
(1/(log_7 ((x/(x−3))))) < (1/(log_7 ((x/3))))  (1) (x/(x−3)) > 0 ∧ (x/3)>0   ⇒ x > 3  (2) log_7 ((x/3)) < log_7 ((x/(x−3)))  (x/3)− (x/(x−3)) < 0  ((x(x−6))/(3(x−3))) < 0 ⇒ x<0 ∨3 < x < 6  the solution ⇒ 3 < x < 6
$$\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{7}} \left(\frac{\mathrm{x}}{\mathrm{x}−\mathrm{3}}\right)}\:<\:\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{7}} \left(\frac{\mathrm{x}}{\mathrm{3}}\right)} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{x}}{\mathrm{x}−\mathrm{3}}\:>\:\mathrm{0}\:\wedge\:\frac{\mathrm{x}}{\mathrm{3}}>\mathrm{0}\: \\ $$$$\Rightarrow\:\mathrm{x}\:>\:\mathrm{3} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{log}_{\mathrm{7}} \left(\frac{\mathrm{x}}{\mathrm{3}}\right)\:<\:\mathrm{log}_{\mathrm{7}} \left(\frac{\mathrm{x}}{\mathrm{x}−\mathrm{3}}\right) \\ $$$$\frac{\mathrm{x}}{\mathrm{3}}−\:\frac{\mathrm{x}}{\mathrm{x}−\mathrm{3}}\:<\:\mathrm{0} \\ $$$$\frac{\mathrm{x}\left(\mathrm{x}−\mathrm{6}\right)}{\mathrm{3}\left(\mathrm{x}−\mathrm{3}\right)}\:<\:\mathrm{0}\:\Rightarrow\:\mathrm{x}<\mathrm{0}\:\vee\mathrm{3}\:<\:\mathrm{x}\:<\:\mathrm{6} \\ $$$$\mathrm{the}\:\mathrm{solution}\:\Rightarrow\:\mathrm{3}\:<\:\mathrm{x}\:<\:\mathrm{6} \\ $$

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