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        riddles >> medium >> logarithm
        
(Message started by: Christine on Jun 1^st, 2012, 10:20am)

Title: logarithm
Post by Christine on Jun 1^st, 2012, 10:20am

How to prove without using calculus

log x <= sqrt (x) for all natural x?

Title: Re: logarithm
Post by pex on Jun 1^st, 2012, 11:03am

How do you define log x without using calculus?

Title: Re: logarithm
Post by Christine on Jun 1^st, 2012, 11:49am

on 06/01/12 at 11:03:01, pex wrote:

How do you define log x without using calculus?

If log x^2 <= sqrt(x^2) then 2*log x <= x
for x positive integer

As for the problem, I thought perhaps we could use induction. Not sure how. I need some help.

Title: Re: logarithm
Post by pex on Jun 1^st, 2012, 12:11pm

on 06/01/12 at 11:49:08, Christine wrote:

If log x^2 <= sqrt(x^2) then 2*log x <= x
for x positive integer

As for the problem, I thought perhaps we could use induction. Not sure how. I need some help.

Let me rephrase. Which properties of the logarithm can we use? For example, you're using log x² = 2 log x, but I think we need calculus to prove that property.

So what I meant to ask was, which definition of the logarithm are you using? Which properties we can use, will follow from that.