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Question Number 5965 by love math last updated on 07/Jun/16

log_(1/2) (x^2 +7x+12)>log_(x+5) (x^2 +7x+12)

$${log}_{\frac{\mathrm{1}}{\mathrm{2}}} \left({x}^{\mathrm{2}} +\mathrm{7}{x}+\mathrm{12}\right)>{log}_{{x}+\mathrm{5}} \left({x}^{\mathrm{2}} +\mathrm{7}{x}+\mathrm{12}\right) \\ $$

Commented byYozzii last updated on 07/Jun/16

((lnu)/(ln0.5))>((lnu)/(ln(x+5)))    (change of base/u=x^2 +7x+12)  lnu((1/(ln0.5))−(1/(ln(x+5))))>0  −−−−−−−−−−−−−−−−−−−−−−−  (1) lnu>0 and (1/(ln0.5))>(1/(ln(x+5)))  u>e^0 =1                       x^2 +7x+12>1  x^2 +7x+11>0  (x+((7+(√5))/2))(x+((7−(√5))/2))>0  ⇒x∈[(−∞,−((7+(√5))/2)≈−4.62)∪((((√5)−7)/2)≈−2.38,+∞)]...(i)    Also, (1/(ln0.5))>(1/(ln(x+5)))  ((ln(x+5)−ln0.5)/((ln0.5)ln(x+5)))>0  ((ln(2(x+5)))/(ln(x+5)))<0  (ln0.5<0)  ⇒(1) ln(2(x+5))>0 & ln(x+5)<0  ⇒2(x+5)>1 & 0<x+5<1  x>−4.5   & −5<x<−4 ⇒x∈(−4.5,−4).  ⇒(2) ln(2(x+5))<0 & ln(x+5)>0  ⇒−5<x<−4.5 & x>−4  (impossible)  ∴ x∈(−4.5,−4)....(ii)  The region of overlap of (i) and (ii)  is empty.  −−−−−−−−−−−−−−−−−−−−−−−−−  lnu<0 and (1/(ln0.5))−(1/(ln(x+5)))<0  lnu<0⇒0<(x+4)(x+3)<1⇒x∈[(−((7+(√5))/2),−4)∪(−3,(((√5)−7)/2))]........ (i)    (1/(ln0.5))−(1/(ln(x+5)))<0⇒((ln(2(x+5)))/(ln(x+5)))>0  ⇒(1) ln(2(x+5))>0 & ln(x+5)>0  x>−4.5  & x>−4⇒x>−4  (2)ln(2(x+5))<0  & ln(x+5)<0  0<2(x+5)<1  &0<x+5<1  −5<x<−4.5  &−5<x<−4⇒ x∈(−5,−4.5)  ∴ x∈[(−4,+∞)∪(−5,−4.5)]....(ii)  Region of overlap of (i) and (ii) is  is x∈[(−((5+(√7))/2),−(9/2))∪(−3,(((√5)−7)/2))].  −−−−−−−−−−−−−−−−−−−−−−−−  Answer: x∈[(((−5−(√7))/2),−(9/2))∪(−3,(((√5)−7)/2))]

$$\frac{{lnu}}{{ln}\mathrm{0}.\mathrm{5}}>\frac{{lnu}}{{ln}\left({x}+\mathrm{5}\right)}\:\:\:\:\left({change}\:{of}\:{base}/{u}={x}^{\mathrm{2}} +\mathrm{7}{x}+\mathrm{12}\right) \\ $$ $${lnu}\left(\frac{\mathrm{1}}{{ln}\mathrm{0}.\mathrm{5}}−\frac{\mathrm{1}}{{ln}\left({x}+\mathrm{5}\right)}\right)>\mathrm{0} \\ $$ $$−−−−−−−−−−−−−−−−−−−−−−− \\ $$ $$\left(\mathrm{1}\right)\:{lnu}>\mathrm{0}\:{and}\:\frac{\mathrm{1}}{{ln}\mathrm{0}.\mathrm{5}}>\frac{\mathrm{1}}{{ln}\left({x}+\mathrm{5}\right)} \\ $$ $${u}>{e}^{\mathrm{0}} =\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$ $${x}^{\mathrm{2}} +\mathrm{7}{x}+\mathrm{12}>\mathrm{1} \\ $$ $${x}^{\mathrm{2}} +\mathrm{7}{x}+\mathrm{11}>\mathrm{0} \\ $$ $$\left({x}+\frac{\mathrm{7}+\sqrt{\mathrm{5}}}{\mathrm{2}}\right)\left({x}+\frac{\mathrm{7}−\sqrt{\mathrm{5}}}{\mathrm{2}}\right)>\mathrm{0} \\ $$ $$\Rightarrow{x}\in\left[\left(−\infty,−\frac{\mathrm{7}+\sqrt{\mathrm{5}}}{\mathrm{2}}\approx−\mathrm{4}.\mathrm{62}\right)\cup\left(\frac{\sqrt{\mathrm{5}}−\mathrm{7}}{\mathrm{2}}\approx−\mathrm{2}.\mathrm{38},+\infty\right)\right]...\left({i}\right) \\ $$ $$ \\ $$ $${Also},\:\frac{\mathrm{1}}{{ln}\mathrm{0}.\mathrm{5}}>\frac{\mathrm{1}}{{ln}\left({x}+\mathrm{5}\right)} \\ $$ $$\frac{{ln}\left({x}+\mathrm{5}\right)−{ln}\mathrm{0}.\mathrm{5}}{\left({ln}\mathrm{0}.\mathrm{5}\right){ln}\left({x}+\mathrm{5}\right)}>\mathrm{0} \\ $$ $$\frac{{ln}\left(\mathrm{2}\left({x}+\mathrm{5}\right)\right)}{{ln}\left({x}+\mathrm{5}\right)}<\mathrm{0}\:\:\left({ln}\mathrm{0}.\mathrm{5}<\mathrm{0}\right) \\ $$ $$\Rightarrow\left(\mathrm{1}\right)\:{ln}\left(\mathrm{2}\left({x}+\mathrm{5}\right)\right)>\mathrm{0}\:\&\:{ln}\left({x}+\mathrm{5}\right)<\mathrm{0} \\ $$ $$\Rightarrow\mathrm{2}\left({x}+\mathrm{5}\right)>\mathrm{1}\:\&\:\mathrm{0}<{x}+\mathrm{5}<\mathrm{1} \\ $$ $${x}>−\mathrm{4}.\mathrm{5}\:\:\:\&\:−\mathrm{5}<{x}<−\mathrm{4}\:\Rightarrow{x}\in\left(−\mathrm{4}.\mathrm{5},−\mathrm{4}\right). \\ $$ $$\Rightarrow\left(\mathrm{2}\right)\:{ln}\left(\mathrm{2}\left({x}+\mathrm{5}\right)\right)<\mathrm{0}\:\&\:{ln}\left({x}+\mathrm{5}\right)>\mathrm{0} \\ $$ $$\Rightarrow−\mathrm{5}<{x}<−\mathrm{4}.\mathrm{5}\:\&\:{x}>−\mathrm{4}\:\:\left({impossible}\right) \\ $$ $$\therefore\:{x}\in\left(−\mathrm{4}.\mathrm{5},−\mathrm{4}\right)....\left({ii}\right) \\ $$ $${The}\:{region}\:{of}\:{overlap}\:{of}\:\left({i}\right)\:{and}\:\left({ii}\right) \\ $$ $${is}\:{empty}. \\ $$ $$−−−−−−−−−−−−−−−−−−−−−−−−− \\ $$ $${lnu}<\mathrm{0}\:{and}\:\frac{\mathrm{1}}{{ln}\mathrm{0}.\mathrm{5}}−\frac{\mathrm{1}}{{ln}\left({x}+\mathrm{5}\right)}<\mathrm{0} \\ $$ $${lnu}<\mathrm{0}\Rightarrow\mathrm{0}<\left({x}+\mathrm{4}\right)\left({x}+\mathrm{3}\right)<\mathrm{1}\Rightarrow{x}\in\left[\left(−\frac{\mathrm{7}+\sqrt{\mathrm{5}}}{\mathrm{2}},−\mathrm{4}\right)\cup\left(−\mathrm{3},\frac{\sqrt{\mathrm{5}}−\mathrm{7}}{\mathrm{2}}\right)\right]........\:\left({i}\right) \\ $$ $$ \\ $$ $$\frac{\mathrm{1}}{{ln}\mathrm{0}.\mathrm{5}}−\frac{\mathrm{1}}{{ln}\left({x}+\mathrm{5}\right)}<\mathrm{0}\Rightarrow\frac{{ln}\left(\mathrm{2}\left({x}+\mathrm{5}\right)\right)}{{ln}\left({x}+\mathrm{5}\right)}>\mathrm{0} \\ $$ $$\Rightarrow\left(\mathrm{1}\right)\:{ln}\left(\mathrm{2}\left({x}+\mathrm{5}\right)\right)>\mathrm{0}\:\&\:{ln}\left({x}+\mathrm{5}\right)>\mathrm{0} \\ $$ $${x}>−\mathrm{4}.\mathrm{5}\:\:\&\:{x}>−\mathrm{4}\Rightarrow{x}>−\mathrm{4} \\ $$ $$\left(\mathrm{2}\right){ln}\left(\mathrm{2}\left({x}+\mathrm{5}\right)\right)<\mathrm{0}\:\:\&\:{ln}\left({x}+\mathrm{5}\right)<\mathrm{0} \\ $$ $$\mathrm{0}<\mathrm{2}\left({x}+\mathrm{5}\right)<\mathrm{1}\:\:\&\mathrm{0}<{x}+\mathrm{5}<\mathrm{1} \\ $$ $$−\mathrm{5}<{x}<−\mathrm{4}.\mathrm{5}\:\:\&−\mathrm{5}<{x}<−\mathrm{4}\Rightarrow\:{x}\in\left(−\mathrm{5},−\mathrm{4}.\mathrm{5}\right) \\ $$ $$\therefore\:{x}\in\left[\left(−\mathrm{4},+\infty\right)\cup\left(−\mathrm{5},−\mathrm{4}.\mathrm{5}\right)\right]....\left({ii}\right) \\ $$ $${Region}\:{of}\:{overlap}\:{of}\:\left({i}\right)\:{and}\:\left({ii}\right)\:{is} \\ $$ $${is}\:{x}\in\left[\left(−\frac{\mathrm{5}+\sqrt{\mathrm{7}}}{\mathrm{2}},−\frac{\mathrm{9}}{\mathrm{2}}\right)\cup\left(−\mathrm{3},\frac{\sqrt{\mathrm{5}}−\mathrm{7}}{\mathrm{2}}\right)\right]. \\ $$ $$−−−−−−−−−−−−−−−−−−−−−−−− \\ $$ $${Answer}:\:{x}\in\left[\left(\frac{−\mathrm{5}−\sqrt{\mathrm{7}}}{\mathrm{2}},−\frac{\mathrm{9}}{\mathrm{2}}\right)\cup\left(−\mathrm{3},\frac{\sqrt{\mathrm{5}}−\mathrm{7}}{\mathrm{2}}\right)\right] \\ $$ $$ \\ $$

Commented bylove math last updated on 07/Jun/16

Then what need to do?

$${Then}\:{what}\:{need}\:{to}\:{do}? \\ $$

Commented byprakash jain last updated on 08/Jun/16

Yozzi considered two cases where  ln (x^2 +7x+12)>0  and second case ln (x^2 +7x+12)<0  range of valid x is also in the answer.

$$\mathrm{Yozzi}\:\mathrm{considered}\:\mathrm{two}\:\mathrm{cases}\:\mathrm{where} \\ $$ $$\mathrm{ln}\:\left({x}^{\mathrm{2}} +\mathrm{7}{x}+\mathrm{12}\right)>\mathrm{0} \\ $$ $$\mathrm{and}\:\mathrm{second}\:\mathrm{case}\:\mathrm{ln}\:\left({x}^{\mathrm{2}} +\mathrm{7}{x}+\mathrm{12}\right)<\mathrm{0} \\ $$ $${range}\:{of}\:{valid}\:{x}\:{is}\:{also}\:{in}\:{the}\:{answer}. \\ $$

Answered by Ashis last updated on 07/Jun/16

log(x+5)>log(1/2)  =>log(2x+10)>0  =>2x+10>1  =>x>−(9/2)

$${log}\left({x}+\mathrm{5}\right)>{log}\left(\mathrm{1}/\mathrm{2}\right) \\ $$ $$=>{log}\left(\mathrm{2}{x}+\mathrm{10}\right)>\mathrm{0} \\ $$ $$=>\mathrm{2}{x}+\mathrm{10}>\mathrm{1} \\ $$ $$=>{x}>−\frac{\mathrm{9}}{\mathrm{2}} \\ $$

Commented byYozzii last updated on 07/Jun/16

What if x=−3? Does the inequality   hold if we assume the logarithm is real valued?

$${What}\:{if}\:{x}=−\mathrm{3}?\:{Does}\:{the}\:{inequality}\: \\ $$ $${hold}\:{if}\:{we}\:{assume}\:{the}\:{logarithm}\:{is}\:{real}\:{valued}?\: \\ $$

Commented byprakash jain last updated on 08/Jun/16

Just to add to Yozzi′s comment.  Complex logarithm and logs of −ve numbers  are complex number and inequality does  not make sense. So real valued logs are  only required.  x∈[−3,−4],x^2 +7x+12≤0 so log cannot be taken.

$$\mathrm{Just}\:\mathrm{to}\:\mathrm{add}\:\mathrm{to}\:\mathrm{Yozzi}'\mathrm{s}\:\mathrm{comment}. \\ $$ $$\mathrm{Complex}\:\mathrm{logarithm}\:\mathrm{and}\:\mathrm{logs}\:\mathrm{of}\:−\mathrm{ve}\:\mathrm{numbers} \\ $$ $$\mathrm{are}\:\mathrm{complex}\:\mathrm{number}\:\mathrm{and}\:\mathrm{inequality}\:\mathrm{does} \\ $$ $$\mathrm{not}\:\mathrm{make}\:\mathrm{sense}.\:\mathrm{So}\:\mathrm{real}\:\mathrm{valued}\:\mathrm{logs}\:\mathrm{are} \\ $$ $$\mathrm{only}\:\mathrm{required}. \\ $$ $${x}\in\left[−\mathrm{3},−\mathrm{4}\right],{x}^{\mathrm{2}} +\mathrm{7}{x}+\mathrm{12}\leqslant\mathrm{0}\:\mathrm{so}\:\mathrm{log}\:\mathrm{cannot}\:\mathrm{be}\:\mathrm{taken}. \\ $$

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