4
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Which number is bigger?

$2^{431},3^{421},4^{321},21^{43},31^{42}$

My attempt:

$4^{321}=2^{642}>2^{431},4^{321}=2^{642}>2^{640}=32^{128}>31^{42}$

$3^{421}>3^{420}=27^{140}>21^43$

But I don't know how to compare $4^{321}$ and $3^{421}$ Any hints?

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  • 1
    $\begingroup$ Are you allowed to assume $log_3 4 $ to a decimal or two? $\endgroup$
    – fleablood
    Jan 11, 2017 at 7:28
  • $\begingroup$ @fleablood No just simple math. $\endgroup$ Jan 11, 2017 at 7:30
  • $\begingroup$ $4^{321}??3^{421}\implies (4/3)^{321}??3^{100}\implies (4/3)^{3.21}??3 $ $4/3^{3.5}=128/27\sqrt {3}<128/27*1.8<2.6 <3$. So $4^{321}< 3^{421} $. There's probably something obvious I am missing. They are close and $4^{321}=8^{214}<9^{210.5}=3^{421} $ because 9>8 is "more important" than 214 > 210.5. But that's intuition. Not proof. $\endgroup$
    – fleablood
    Jan 11, 2017 at 8:09

1 Answer 1

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$4^{321}=2^{642}=2^{11*58+4}=16*2048^{58}<27*2187^{58}=3^3*3^{7*58}=3^{409}<3^{421}$

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