The ability of manually comparing any powers of any natural numbers is one of my greatest dreams. "Manually" means without a calculator, or a computer, or any other device besides my own head and possibly also a pencil and paper.
For example, we need to compare 17⁹⁷ and 63⁶⁴.
In a video published on Youtube I learnt an interesting algorithm.
17 > 16. Thus, 17⁹⁷ > 16⁹⁷ = (2⁴)⁹⁷ = 2³⁸⁸.
63 < 64. Thus, 63⁶⁴ < 64⁶⁴ = (2⁶)⁶⁴ = 2³⁸⁴.
So, finally, 17⁹⁷ > 63⁶⁴.
https://www.youtube.com/watch?v=lpZZ6DSVI3o
This is very interesting and instructive, but I feel that it may not work in some cases. E.g. we must compare 12⁶⁷ and 8⁷⁷. I fear that the above mentioned algorithm will fail.
12 > 8. Thus, 12⁶⁷ > 8⁶⁷ = (2³)⁶⁷ = 2²⁰¹. Or, on the other hand, 12 < 16. Thus, 12⁶⁷ < 16⁶⁷ = (2⁴)⁶⁷ = 2²⁶⁸. And 8⁷⁷ = (2³)⁷⁷ = 2²³¹.
So we now know that 12⁶⁷ lies between 2²⁰¹ and 2²⁶⁸, but it is unclear if it is greater or less than 2²³¹ = 8⁷⁷.
And as of taking the logarithms... Yes, it solves the problem, of course. I know that any logarithm with a base greater than 1 is a strictly increasing function. So instead of direct comparison of powers we can take either decimal or natural logarithms of both our numbers (the exponent will become a factor) and then compare the resulting logarithms with each other. But... my question is about how to manually compare the powers. And unfortunately I do not know how to take an arbitrary logarithm without usage of a calculator.
So is there a universal algorithm which works in every single case? (Or, if it isn't, is there a way to somehow prove that it doesn't exist?)