Math: Which One Is Bigger, 1000^1001 or 1001^1000?

The question looks like this:

1000¹⁰⁰¹ (?) 1001¹⁰⁰⁰

We want to find the (?) operator.

I scribbled that a moment ago. I'm using logarithm, and captured the typed-steps as an image.

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Let's start

Hang on for a moment...

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In conclusion

Without using calculator,
1000¹⁰⁰¹ is larger than 1001¹⁰⁰⁰

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10001001 > 10011000

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Closure
As I observed, that pattern starts from integer greater than 2 (to any large positive integer).
I haven't tinkered the positive "infinity". I suppose it will be the same.
It's like imagination, the infinity concept.

And for number with, like, floating point, you can do your own scribbling.

Anyway, if we use 1 (in that pattern above):

1² (?) 2¹
1 (?) 2
1 < 2

If we use 2:

2³ (?) 3²
8 (?) 9
8 < 9

The next one, 3:

3⁴ (?) 4³
81 (?) 64
81 > 64

Next, 4:

4⁵ (?) 5⁴
1024 (?) 625
1024 > 625

And so on...

This technique will surely result different outcome for integer less than or equal to 2. Especially on the Let's assume it's 1 step (rounding down step).

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Wolfram and Wikipedia

This is the link on Wolfram|Alpha about that particular question.

This is the the form equivalency and decimal approximation of log₁₀₀₀(1001) on Wolfram|Alpha.

This is the list of logarithmic identities on Wikipedia.