Prove that ln (((√(13))−1)/(10)) + (√(13)) − 2 >0 without calculator.


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Question Number 212098 by CrispyXYZ last updated on 30/Sep/24

$Prove that$ $\ln \frac{\sqrt{13} - 1}{10} + \sqrt{13} - 2 > 0$ $without calculator .$
	Answered by MrGaster last updated on 03/Nov/24

	$\ln (\sqrt{13} - 1) - \ln 10 + \sqrt{13} - 2 > 0$ $\ln (\sqrt{13} - 1) + \sqrt{13} > 2$ $in consideration of \sqrt{13} < 3 (∵ e^{3} > 10), ∴ :$ $\ln (\sqrt{13} - 1) + \sqrt{13} > \ln (3 - 1) + 3$ $> \ln 2 + 3$ $> 1 + 2$ $> 2 + \ln 10$ $∴,$ $\ln \frac{\sqrt{13} - 1}{10} + \sqrt{13} - 2 > 0$
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