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?
Which number is bigger than the others?
4
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algebra-precalculus
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1Are you allowed to assume $log_3 4 $ to a decimal or two? – 2017-01-11
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0@fleablood No just simple math. – 2017-01-11
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0$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. – 2017-01-11
1 Answers
6
$4^{321}=2^{642}=2^{11*58+4}=16*2048^{58}<27*2187^{58}=3^3*3^{7*58}=3^{409}<3^{421}$