Which-one-is-largest-without-using-calculator-31-11-or-17-14-

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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