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[SOLVED] Prove 1/(log₂π)+1/(log₅π)>2

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,601
Prove $\dfrac{1}{\log_2 \pi}+\dfrac{1}{\log_5 \pi}>2 $.
 

kaliprasad

Well-known member
Mar 31, 2013
1,283
$\frac{1}{\log_2\pi } +\frac{1}{\log_5\pi } $
= $\log_{\pi} 2 +\log_{\pi} 5$ using $\log_a b * \log_b a = 1$
= $\log_{\pi} 10$

Now $\pi = 3.14 < 3.15$
so $\pi^2 < 31.5^2$ or $\pi^2 < 992.25$ as $31 * 32 = 992$
so $ 10 > \pi^2$
so we have
$\log_{\pi} 10 > 2$ and hence the result
 
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