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

The question looks like this:

$1000^1001 (?) 1001^1000$

We want to find the (?) operator.

I scribbled that using logarithmic, here it goes.

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Scribble

$1000^1001 (?) 1001^1000$

Let us assume:

$1000 = a$

Therefore:

$1001 = (a + 1)$

So this:

$1000^1001 (?) 1001^1000$

Becomes:

$a^(a+1) (?) (a + 1)^a$

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➡️ We then place log (logarithmic function) on both sides.

This form:

$a^(a+1) (?) (a + 1)^a$

➡️ Becomes:

$log_a(a^(a+1)) (?) log_a((a + 1)^a)$

Continue by implementing logarithmic properties.

The form above can be transformed into this:

$(a + 1) log_a(a) (?) (a) log_a(a + 1)$

We can simplify $log_a(a) = 1$. Thus:

$(a + 1) (1) (?) (a) log_a(a + 1)$

$log_a(a + 1) > 1$ — very near 1 because $a = 1000$.

Therefore we can safely simplify that $log_a(a + 1) = 1$

Final simplification:

➡️ $(a + 1) (1) (?) (a) (1)$

$(a + 1) (?) (a)$

💡 For any number, adding one gives a result larger than the number itself — naturally. Thus:

$(a + 1) > a$

Meaning:

$1000^1001$ is greater than $1001^1000$

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$1000^1001 > 1001^1000$

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The operator for the answer is >

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Pattern

As I observed, the pattern starts from integer greater than 2 (to any large positive integer).

Let's try 1:

1² (?) 2¹
1 (?) 2
1 < 2

2:

2³ (?) 3²
8 (?) 9
8 < 9

Next, 3:

3⁴ (?) 4³
81 (?) 64
81 > 64

4:

4⁵ (?) 5⁴
1024 (?) 625
1024 > 625

5:

5⁶ (?) 6⁵
15625 (?) 7776
15625 > 7776

So Forth...

It's like that, the pattern I observed.

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