Math-puzzle: How to show that 17^14 > 31^11?

[Guest]

Hi!

I know that the power of 17^14 is greater, but I do wonder how you could show it without using obviously any calculation help, but with the mere use of the laws of potency?
I just cannot find even a start and would appreciate any input and any hint.

regards,
Placebo

[Guest]

hi

17¹⁴==17¹¹*17³

if you show:
17¹¹*17³ { } 31¹¹
and it is equivalent whit 17³ { } 31¹¹ / 17¹¹
and this 31¹¹ / 17¹¹=(31/17)¹¹ [tex]\approx[/tex] 2¹¹

now you can compare 17³ { } 2¹¹
and this you can easy comare

17³{ > }2¹¹

[Placebo]

Hey,
has helped a lot - thanks!:)

[Guest]

[tex]17^{14}> 31^{11}[/tex] if and only if
[tex]14log(17)> 11log(31)[/tex] which is true if and only if [tex]\frac{14}{11}> \frac{log(31)}{log(17)}[/tex].

[tex]\frac{14}{11}= 1.272727272727272...[/tex]
and
[tex]\frac{log(31)}{log(17}= 1.21204681309626[/tex].