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#1 2006-03-09 22:54:27

ganesh
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Logarithms

L # 1

Show that


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#2 2006-03-10 01:41:40

krassi_holmz
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Re: Logarithms


So:

Last edited by krassi_holmz (2006-03-10 01:44:53)


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#3 2006-03-10 02:01:49

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Re: Logarithms

http://kaneart.metadogs.com/images/new_index_images/very_good.jpg


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#4 2006-03-10 02:06:00

ganesh
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Re: Logarithms

L # 2

If


show that xyz=1.


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#5 2006-03-10 06:19:45

krassi_holmz
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Re: Logarithms

Let
log x = x'
log y = y'
log z = z'.
Then:



and


Now, consider that from xyz=1 follows:


But x'=log x; y'=log y' z'=log z, so we must prove that:

x'+y'+z'=0.

Rewriting in terms of x' gives:


q.E.d.


IPBLE:  Increasing Performance By Lowering Expectations.
 

#6 2006-03-10 15:17:14

ganesh
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Re: Logarithms

Well done, krassi_holmz!


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#7 2006-03-11 15:48:02

ganesh
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Re: Logarithms

L # 3

If xy=a and log (x/y)=b, then what is the value of (logx)/(logy)?


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#8 2006-03-11 19:03:38

krassi_holmz
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Re: Logarithms

loga=2logx+3logy
b=logx-logy
loga+3b=5logx
loga-2b=3logy+2logy=5logy
logx/logy=(loga+3b)/(loga-2b).

Last edited by krassi_holmz (2006-03-11 19:06:29)


IPBLE:  Increasing Performance By Lowering Expectations.
 

#9 2006-03-11 19:23:36

ganesh
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Re: Logarithms

Very well done, krassi_holmz! up


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#10 2009-01-05 10:58:47

JaneFairfax
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Re: Logarithms

L # 4


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#11 2009-01-05 15:13:57

ganesh
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Re: Logarithms


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#12 2009-01-05 22:37:30

JaneFairfax
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Re: Logarithms

You are not supposed to use a calculator or log tables for L # 4. shame Try again!

Last edited by JaneFairfax (2009-01-05 22:40:20)


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A: Click here for answer.
 

#13 2009-01-06 00:36:33

ganesh
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Re: Logarithms

No, I didn't cool
I remember


and
.


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#14 2009-01-06 20:57:49

JaneFairfax
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Re: Logarithms

You still used a calculator / log table in the past to get those figures (or someone else did and showed them to you). I say again: no calculators or log tables to be used (directly or indirectly) at all!! neutral

Last edited by JaneFairfax (2009-01-06 23:30:04)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#15 2009-02-07 22:31:40

JaneFairfax
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Re: Logarithms


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#16 2010-04-19 17:06:39

keveenjones
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Re: Logarithms

log a = 2log x + 3log y

b = log x log y

log a + 3 b = 5log x

loga - 2b = 3logy + 2logy = 5logy

logx / logy = (loga+3b) / (loga-2b)

 

#17 2010-04-19 19:04:41

rzaidan
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Re: Logarithms

Hi ganesh
for  L # 1
since log(a)= 1 / log(b),    log(a)=1
            b               a            a
we have
1/log(abc)+1/log(abc)+1/log(abc)=
       a                 b                c
log(a)+log(b)+log(c)= log(abc)=1
abc      abc          abc    abc
Best Regards
Riad Zaidan

 

#18 2010-04-19 19:14:13

rzaidan
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Re: Logarithms

Hi ganesh
for  L # 2
I think that  the following proof  is easier:
Assume Log(x)/(b-c)=Log(y)/(c-a)=Log(z)/(a-b)=t
So Log(x)=t(b-c),Log(y)=t(c-a)  ,  Log(z)=t(a-b) So Log(x)+Log(y)+Log(z)=tb-tc+tc-ta+ta-tb=0
So Log(xyz)=0 so   xyz=1   Q.E.D
Best Regards
Riad Zaidan

 

#19 2010-04-19 23:02:18

ganesh
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Re: Logarithms

Gentleman,

Thanks for the proofs.
Regards.


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#20 2010-08-17 14:15:11

jonnyj99
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Re: Logarithms

log_2(16) = \log_2 \left ( \frac{64}{4} \right ) = \log_2(64) - \log_2(4) = 6 - 2 = 4, \,

log_2(\sqrt[3]4) = \frac {1}{3} \log_2 (4) = \frac {2}{3}. \,

 

#21 2011-05-28 17:59:31

reconsideryouranswer
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Re: Logarithms

JaneFairfax wrote:

L # 4

I don't want a method that will rely on defining certain functions, taking derivatives,
noting concavity, etc.





Change of base:






Each side is positive, and multiplying by the positive denominator
keeps whatever direction of the alleged inequality the same direction:



On the right-hand side, the first factor is equal to a positive number less than 1,
while the second factor is equal to a positive number greater than 1.  These
facts are by inspection combined with the nature of exponents/logarithms.

Because of (log A)B = B(log A) = log(A^B), I may turn this into:



I need to show that


Then


Then 1 (on the left-hand side) will be greater than the value on the
right-hand side, and the truth of the original inequality will be established.

I want to show


Raise a base of 3 to each side:



Each side is positive, and I can square each side:





-----------------------------------------------------------------------------------

Then I want to show that when 2 is raised to a number equal to
(or less than) 1.5, then it is less than 3.



Each side is positive, and I can square each side:









Last edited by reconsideryouranswer (2011-05-28 18:05:01)


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#22 2011-05-28 21:57:00

ganesh
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Re: Logarithms

Hi reconsideryouranswer,

This problem was posted by JaneFairfax. I think it would be appropriate she verify the solution.


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