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Algebra Level 2

Which one is greater?

$\large {1000^{1001} \text{ OR } 1001^{1000}}$

Can't be determined Both are equal

1001^{1000}

1000^{1001}

2 solutions

Dev Sharma
Aug 17, 2015

1000^1001 has 3003 digits.

1001^1000 has 3001 digits.

Thus 1000^1001 is greater.

$1000^{1001}>1001^{1000}\iff 1000^{1/1000}>1001^{1/1001}$ , which is true, since

$x^{1/x}$ is strictly decreasing in $(e,+\infty)$ , since

$(x^{1/x})'=-x^{1/x-2} (\log(x)-1)<0$ for $x>e$ .

mathh mathh - 5 years, 10 months ago

$1000^{1001}$ has 3004 digits, you forgot to add one in the characteristic of $log\ (1000^{1001}$ )

jaiveer shekhawat - 5 years, 10 months ago

how do you find the no. of digits for 1001^1000

Anupam Shandilya - 5 years, 10 months ago

$\mbox{number of digits} = \left \lfloor \log n \right \rfloor+1$ where $n$ is a positive integer.

Jaydee Lucero - 5 years, 9 months ago

but how do we find log 1001 without log tables

Anupam Shandilya - 5 years, 9 months ago

You are all wrong

Roniel Hernandez - 5 years, 9 months ago

Jaiveer Shekhawat
Aug 18, 2015