Which one is largest? (without using calculator) 31^(11) or 17^(14) ??


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Question Number 11048 by Joel576 last updated on 09/Mar/17

$Which one is largest ? (without using calculator)$ $31^{11} or 17^{14} ? ?$
	Answered by ajfour last updated on 09/Mar/17

	Answered by ajfour last updated on 09/Mar/17

	Commented by ajfour last updated on 09/Mar/17

	$s o 17^{14} s h o u l d b e g r e a t e r .$
	Commented by Joel576 last updated on 10/Mar/17

	$t h a n k y o u v e r y m u c h$
	Answered by ajfour last updated on 09/Mar/17

	$\log (31^{11)} = 11 (\log 31)$ $= 11 (1 + \log 3.1) = 11 (1 + 0.48)$ $= 11 \times 1.48 = 14.8 + 1.48 = 16.28$ $w h i l e \log (17^{14}) = 14 (\log 17)$ $\approx 14 \times 4 (\log 2) = 56 \times 0.302 \approx 17$ $t h u s 17^{14} s h o u l d b e g r e a t e r .$
	Answered by mrW1 last updated on 09/Mar/17

	$w i t h o u t u s i n g c a l c u l a t o r :$ $\log_{2} (31^{11}) = 11 \times \log_{2} 31$ $< 11 \times \log_{2} 32 = 11 \times \log_{2} 2^{5} = 11 \times 5 = 55$ $\log_{2} (17^{14}) = 14 \times \log_{2} 17$ $> 14 \times \log_{2} 16 = 14 \times \log_{2} 2^{4} = 14 \times 4 = 56$ $∴ \log_{2} (17^{14}) > 56 > 55 > \log_{2} (31^{11})$ $∵ 17^{14} > 31^{11}$
	Commented by Joel576 last updated on 10/Mar/17

	$t h a n k y o u v e r y m u c h$
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