Order of Huge Numbers

Algebra Level 2

$\large \boxed{ 111^{110} , \quad 110^{111} , \quad 109^{112} , \quad 112^{109} }$

Which option is correct about the order of the above numbers?

109^{112} > 110^{111} > 111^{110} > 112^{109}

109^{112} > 111^{110} > 110^{111} > 112^{109}

111^{110} > 110^{111} > 109^{112} > 112^{109}

111^{110} > 112^{109} > 109^{112} > 110^{111}

2 solutions

Surya Sharma
Feb 15, 2016

Find the number of digits in their expansion by using logarithms /(111^110/) has 225 digits /(110^111/) has 227 digits /(109^112/) has 229 digits /(112^109/) has 224 digits And hence the answer

Otto Bretscher
Feb 15, 2016

$f(x)=x^{221-x}$ is decreasing on $[109,112]$ since $f'(x)=-x^{220-x}(x+x\ln(x)-221)<0$ . Note that $x\ln(x)>400$ on the given interval.

Order of Huge Numbers

2 solutions

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