Which is larger: 1000^1001 or 1001^1000

Here is a proof without a calculator:

Start out by letting:
x = 1000^1001 and y = 1001^1000.

By taking the natural logarithm of both sides:
ln(x) = ln(1000^1001) and ln(y) = ln(1001^1000)
==> ln(x) = 1001ln(1000) and ln(y) = 1000ln(1001).

Since ln(x) > ln(y) implies x > y and ln(x) < ln(y) implies that x < y as ln(x) is an increasing function, we want to see which is bigger: ln(x) or ln(y).

To do this, note that f(x) = x/ln(x) is an increasing function for x > e.
(You can prove this by noting that f'(x) = [ln(x) - 1]/x^2 > 0 for x > e.)

Since 1001 > 1000, we see that:
f(1001) > f(1000) ==> 1001/ln(1001) > 1000/ln(1000)

Multiplying both sides of the inequality by ln(1000)ln(1001) yields:
10001ln(1000) > 1000ln(1001) ==> ln(x) > ln(y).

Therefore, x > y and 1000^1001 is greater.

I hope this helps!

The first one, it multiples 1000 one more time than the other one, which just adds a one at the end, rather than multiplying by 1000 again.

first one 1000^1001 because it adds another 'ones' place to the final answer, while 1001^1000 only changes the answer

1000^1001 = 1e3003

1001^1000 = 2.717e3000

The first is larger.

the first one. there will be 1001 "0's" added behind vs 1000 "0's" if you want to make it similar but smaller then it would be like 10,000,000 vs 1,001,000

1000^1001

the second one