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# Proving an q^n is bigger or smaller than m^p watch

1. Hello

Is there an efficient way of finding whichever one of 218 or 515 is bigger without a calculator, and not by multiplying until you get the value of the two?
2. Instinct tells me to start from

21^8 < 25^8

But I'm almost certainly wrong.

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3. Although I'm sure someone can come up with something better, I'll give it a shot.

Attempt one (long):

Spoiler:
Show

Lets say we have and

To directly compare them, one easily method would to convert say, q, into the form

So to do this;

Making;

This can be expanded out using

Thus,

Now we have them both as powers of m, so just compare the powers.

In your example and , if we use the same method, and put 21 as a power of 5.

Attempt 2 (same method, shorter). I was naive in not just considering this:

Spoiler:
Show

and

If

Or if:

Obviously this requires some rough calculation of logs, however I'm skeptical there is a method which involves next to no calculations. Considering the numbers could take any value, and just because the power of one is bigger, it doesn't necessarily mean the number is bigger.

Edit- Found this: http://www.zachwg.org/logarithms.pdf, could be useful.
4. (Original post by Krollo)
Instinct tells me to start from

21^8 < 25^8

But I'm almost certainly wrong.
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lol
5. (Original post by Phichi)
x
I can see where you are going with method one and I understand the basis of your workings, but when it comes to logarithms I clearly need to study it more (I'm in the AS year).

Thanks for sharing, I will try and seek the knowledge you have demonstrated.
6. (Original post by Krollo)
Instinct tells me to start from

21^8 < 25^8

But I'm almost certainly wrong.

Posted from TSR Mobile
You are right that you are wrong but it's always good to try.
I know a step I won't be trying now :P
7. (Original post by Wunderbarr)
You are right that you are wrong but it's always good to try.
I know a step I won't be trying now :P
Lol. For some reason my intuition often fails me for maths problems like this.

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8. (Original post by Wunderbarr)
Hello

Is there an efficient way of finding whichever one of 218 or 515 is bigger without a calculator, and not by multiplying until you get the value of the two?
That's vs ; so we want vs . The quotient is about , which if you can be bothered to find which is roughly . Hence the quotient is greater than 1, so ; hence is bigger.

That only required finding and approximating fairly cavalierly.

Alternatively you can say which is . You can work out 0.84^8 by repeated squaring if you want.

Neither of those ways are nice, sadly.

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Updated: December 24, 2014
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