Comparing large exponents.

Question

I have come up with a way to compare large exponents, for example: I can tell which number is bigger in $12345^{78901}$ or $21346^{78900}$ within a few seconds without using calculator. So I have 2 questions.

1)Is it something important?(I dont know if it something simple that anyone can do) Can you compare them easily?

2)If it is something useful how and where can I publish it?

(I didn't post it on academia stack exchange because of the first question)

Update:

$9873^{64}$ > $11424.9^{63}$(took me about 5 seconds to compare , using only pen and paper with no calculator)

I don't know if this was a good example but I can do the same for even bigger numbers in which the answers are very close to each other. Seeing the comments I think it is a pretty good method, so how do I let people in masses to know about it. It is not a research paper but just a simple method so, is there any magazine that publishes this kind of stuff?

Answer 1

The method you proposed in your (now deleted) post is not valid. It is simply not true that $$a>b\implies x^a>y^b \quad \forall x, y$$

Specifically, you err when you assert that $$\frac {\log_c A}{\log_c B}=\log_c (A-B)$$

To see the difficulty of the problem, note that, while it is true that $$10375^{105}> 70288^{87}$$ we also have $$10375^{105}< 70289^{87}$$

Thus, whatever method you use it must be sensitive enough to detect the difference between $70288$ and $70289$. That's why I suggested that this would be a difficult test case.

Answer 2

You can break this problem down using some pretty gross estimations.

e.g. $21346 > 1.5\cdot 12345 > \sqrt 2\cdot 12345$

$21346^{78900} > \sqrt 2^{78900} \cdot 12345^{78900}$

$12345^{78901} = 12345\cdot 12345^{78900}$

If $\sqrt 2^{78900} > 12345$ we are done.

$2^{20} \approx 1$ million

$\sqrt 2^{40} > 12345$

$21346^{78900}$ is significantly bigger than $12345^{78901}$