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Question Number 212098 by CrispyXYZ last updated on 30/Sep/24
Prove that  ln (((√(13))−1)/(10)) + (√(13)) − 2 >0  without calculator.
$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{ln}\:\frac{\sqrt{\mathrm{13}}−\mathrm{1}}{\mathrm{10}}\:+\:\sqrt{\mathrm{13}}\:−\:\mathrm{2}\:>\mathrm{0} \\ $$$$\mathrm{without}\:\mathrm{calculator}. \\ $$
Answered by MrGaster last updated on 03/Nov/24
ln((√(13))−1)−ln 10+(√(13))−2>0  ln((√(13))−1)+(√(13))>2  in consideration of (√(13))<3(∵e^3 >10),∴ :  ln((√(13))−1)+(√(13))>ln(3−1)+3  >ln2+3  >1+2  >2+ln 10  ∴,  ln(((√(13))−1)/(10))+(√(13))−2>0
$$\mathrm{ln}\left(\sqrt{\mathrm{13}}−\mathrm{1}\right)−\mathrm{ln}\:\mathrm{10}+\sqrt{\mathrm{13}}−\mathrm{2}>\mathrm{0} \\ $$$$\mathrm{ln}\left(\sqrt{\mathrm{13}}−\mathrm{1}\right)+\sqrt{\mathrm{13}}>\mathrm{2} \\ $$$$\mathrm{in}\:\mathrm{consideration}\:\mathrm{of}\:\sqrt{\mathrm{13}}<\mathrm{3}\left(\because{e}^{\mathrm{3}} >\mathrm{10}\right),\therefore\:: \\ $$$$\mathrm{ln}\left(\sqrt{\mathrm{13}}−\mathrm{1}\right)+\sqrt{\mathrm{13}}>\mathrm{ln}\left(\mathrm{3}−\mathrm{1}\right)+\mathrm{3} \\ $$$$>\mathrm{ln2}+\mathrm{3} \\ $$$$>\mathrm{1}+\mathrm{2} \\ $$$$>\mathrm{2}+\mathrm{ln}\:\mathrm{10} \\ $$$$\therefore, \\ $$$$\mathrm{ln}\frac{\sqrt{\mathrm{13}}−\mathrm{1}}{\mathrm{10}}+\sqrt{\mathrm{13}}−\mathrm{2}>\mathrm{0} \\ $$

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