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3-log-x-2-log-x-2-log-x-4-gt-0-




Question Number 87773 by john santu last updated on 06/Apr/20
^3 log (^x^2  log (^x^2  log x^4 ))> 0
$$\:^{\mathrm{3}} \mathrm{log}\:\left(\:^{\mathrm{x}^{\mathrm{2}} } \mathrm{log}\:\left(\:^{\mathrm{x}^{\mathrm{2}} } \mathrm{log}\:\mathrm{x}^{\mathrm{4}} \right)\right)>\:\mathrm{0} \\ $$
Commented by john santu last updated on 06/Apr/20
⇒^x^2  log (^x^2  log x^4 ) > 1  (i)x ≠ 0  (ii) ^x^2  log (^x^2  log x^4 ) >^x^2  log x^2   ⇒(x^2 −1)(x^4 −x^2 ) > 0  (x+1)^2  (x−1)^2  x^2  >0   x≠ −1 , x ≠ 1   (iii) 1 < x^2  < 2  { ((−(√2) < x < −1)),((1 < x < (√2))) :}
$$\Rightarrow\:^{\mathrm{x}^{\mathrm{2}} } \mathrm{log}\:\left(\:^{\mathrm{x}^{\mathrm{2}} } \mathrm{log}\:\mathrm{x}^{\mathrm{4}} \right)\:>\:\mathrm{1} \\ $$$$\left(\mathrm{i}\right)\mathrm{x}\:\neq\:\mathrm{0} \\ $$$$\left(\mathrm{ii}\right)\:\:^{\mathrm{x}^{\mathrm{2}} } \mathrm{log}\:\left(\:^{\mathrm{x}^{\mathrm{2}} } \mathrm{log}\:\mathrm{x}^{\mathrm{4}} \right)\:>\:^{\mathrm{x}^{\mathrm{2}} } \mathrm{log}\:\mathrm{x}^{\mathrm{2}} \\ $$$$\Rightarrow\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{4}} −\mathrm{x}^{\mathrm{2}} \right)\:>\:\mathrm{0} \\ $$$$\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} \:\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{2}} \:\mathrm{x}^{\mathrm{2}} \:>\mathrm{0}\: \\ $$$$\mathrm{x}\neq\:−\mathrm{1}\:,\:\mathrm{x}\:\neq\:\mathrm{1}\: \\ $$$$\left(\mathrm{iii}\right)\:\mathrm{1}\:<\:\mathrm{x}^{\mathrm{2}} \:<\:\mathrm{2}\:\begin{cases}{−\sqrt{\mathrm{2}}\:<\:\mathrm{x}\:<\:−\mathrm{1}}\\{\mathrm{1}\:<\:\mathrm{x}\:<\:\sqrt{\mathrm{2}}}\end{cases} \\ $$
Commented by mahdi last updated on 06/Apr/20
what meaning  (x^2 ) in     ^x^2  log ?
$$\mathrm{what}\:\mathrm{meaning}\:\:\left(\mathrm{x}^{\mathrm{2}} \right)\:\mathrm{in}\:\:\:\:\overset{\mathrm{x}^{\mathrm{2}} } {\:}\mathrm{log}\:? \\ $$$$ \\ $$
Commented by john santu last updated on 06/Apr/20
it same to log_x^2   (x^4 )
$$\mathrm{it}\:\mathrm{same}\:\mathrm{to}\:\mathrm{log}_{\mathrm{x}^{\mathrm{2}} } \:\left(\mathrm{x}^{\mathrm{4}} \right) \\ $$
Commented by mahdi last updated on 06/Apr/20
thanks
$$\mathrm{thanks} \\ $$

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