logarithms

["Luis Lopes" <llopes@xxxxxxxxxxxxx>]

Thank you.
Lui's

-----Mensagem Original-----
De: <ccrousse@memphis.edu>
Para: Luis Lopes <llopes@ensrbr.com.br>
Enviada em: Segunda-feira, 7 de Agosto de 2000 12:24
Assunto: Re: our list


> Dear Luis:
>
>    I'm afraid that I really don't have a very satisfying, elementary
> proof of the problem with the logarithms, but here goes.  The given
> equation yields
>
> 1 + x^{1/2} = 2^{\log_3 x} = x^a   where a  = log 2/log 3> 1/2.
>
> Writing this equation as  F(x) = 1 + x^{1/2} - x^a = 0, it is clear that
any
> solution must satisfy x > 1.  Note that F'(x) = (1/2)x^{-1/2} - a x^{a-1}
< 0
> for z > 1 so F decreases monotonically on (1, \infty).  F(1) = 1 and F(x)
< 0
> for
> all sufficiently large x, so there is a unique solution of the equation
> F(x) = 0.  By inspection, this unique solution is x = 9,
>
> Cecil
>
>
>
> Luis Lopes wrote:
>
> > Dear Cecil,
> >
> > >    Is it possible that the second problem actually reads
> > >
> > >  \left\lfloor \frac{m^2 x-13}{1999} \right\rfloor = \frac{x-12}{2000}
?
> >
> > You are right. I browsed through the moldavian site and saw the
> > original text. Incidentally all the last three problems that you solved
> > (inequality, constant function, and this one) come from the same site.
> >
> > Your solution follows:
> >
> > > For m = \pm 1 this equation has 1999 solutions, namely
> > >
> > > x = 2000k + 12,   k = 1,2,\ldots,1999.
> > >
> > I raise my concerns to problems such as this: how can one solve this
> > kind of problem? Does one have to make a guess and confirm it is right?
> > How about a systematic way of thinking, ie, brutal force?
> >
> > Look at this easier problem, taken from the same site: solve the
equation
> >
> > \log_2 ( 1 + \sqrt{x} ) = \log_3 x
> >
> > I tried brutal force and no thinking and got nothing. But one sees
(guesses)
> > almost immediately that x=9 is a solution. Should this be considered a
true
> > solution in a high school exam? I would say yes but ...
> > How about other solutions? I still have to check that.
> >
> > Anyway, I am thinking loudly.
> >
> > I propose now another problem from the list:
> >
> > Let a_n = 6^n + 8^n. Calculate the remainder of \frac{a_{93}}{a_{49}} .
> >
> > The email of the list is
> >
> > Obm-l@mat.puc-rio.br
> >
> > To subscribe, send a message to
> >
> > Majordomo@mat.puc-rio.br
> >
> > In the body, write   Subscribe obm-l
> >
> > Hope it works easily.
> >
> > Best regards,
> > Luis