How would I find which of 3^303 and 2^454 is bigger?

Question

ganeshie8 · Answer

Are you allowed to use logarithms here?

zyberg · Answer

Not really, but I would be interested in seeing a solution with logarithms as well!

ganeshie8 · Answer

You may need calculator if you use logarithms.  Are you okay with that?

zyberg · Answer

Yes.

zyberg · Answer

(Although, if anyone have ideas how to solve this without calculator, I would appreciate seeing them :) )

ganeshie8 · Answer

Then simply check if 2^(303/454) is greater than 3

ganeshie8 · Answer

Your calculator shouldn't have any problem computing that

zyberg · Answer

I had an idea similar to yours, however, I wonder, if it's possible in any way to solve this without calculator?

ganeshie8 · Answer

Awesome! I'm wondering the same. Let's wait for ideas from others...

ganeshie8 · Answer

Typo fixed :
Then simply check if 2^(454/303) is greater than 3

zyberg · Answer

Just found an interesting way to solve this without calculator, but with use of logarithms.
since $$\log_{2}(3) = x; \log_{2}(3^{10})=10x; 3^{10}= 59049; 2^{15}=3276; 2^{16}= 65536; 15 < 10x < 16; 1.5 < x < 1.6$$$$ 2^{15}=3276; 2^{16}= 65536; 15 < 10x < 16; 1.5 < x < 1.6$$
So, 454/303 must be equal more than 1.6 for 2^454 to be more than 3^303. But 454/303 = 1498... So, 3^303 is a bigger number.

This way it's possible to do everything without calculator (although a little hassle, but well..) Thanks for getting me on the way, @ganeshie8

sachintha · Answer

Great job! @Zyberg  :)

zyberg · Answer

Thank you, @Sachintha ;)

ganeshie8 · Answer

Fantastic! I can't give you a medal as I'm on my mobile atm. I'll give it once I get on my computer :)

zyberg · Answer

@ganeshie8 thank you :)