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#1 2006-03-08 23:54:27

ganesh
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Registered: 2005-06-28
Posts: 37,090

Logarithms

L # 1

Show that


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#2 2006-03-09 02:41:40

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Logarithms


So:

Last edited by krassi_holmz (2006-03-09 02:44:53)


IPBLE:  Increasing Performance By Lowering Expectations.

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#3 2006-03-09 03:01:49

ganesh
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Registered: 2005-06-28
Posts: 37,090

Re: Logarithms

very_good.jpg


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#4 2006-03-09 03:06:00

ganesh
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Registered: 2005-06-28
Posts: 37,090

Re: Logarithms

L # 2

If


show that xyz=1.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#5 2006-03-09 07:19:45

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

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.

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#6 2006-03-09 16:17:14

ganesh
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Registered: 2005-06-28
Posts: 37,090

Re: Logarithms

Well done, krassi_holmz!


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#7 2006-03-10 16:48:02

ganesh
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Registered: 2005-06-28
Posts: 37,090

Re: Logarithms

L # 3

If x²y³=a and log (x/y)=b, then what is the value of (logx)/(logy)?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#8 2006-03-10 20:03:38

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

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 (2006-03-10 20:06:29)


IPBLE:  Increasing Performance By Lowering Expectations.

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#9 2006-03-10 20:23:36

ganesh
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Registered: 2005-06-28
Posts: 37,090

Re: Logarithms

Very well done, krassi_holmz! up


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#10 2009-01-04 11:58:47

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Logarithms

L # 4

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#11 2009-01-04 16:13:57

ganesh
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Registered: 2005-06-28
Posts: 37,090

Re: Logarithms


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#12 2009-01-04 23:37:30

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Logarithms

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

Last edited by JaneFairfax (2009-01-04 23:40:20)

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#13 2009-01-05 01:36:33

ganesh
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Registered: 2005-06-28
Posts: 37,090

Re: Logarithms

No, I didn't cool
I remember


and
.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#14 2009-01-05 21:57:49

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

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 00:30:04)

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#15 2009-02-06 23:31:40

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Logarithms

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#16 2010-04-18 19:06:39

keveenjones
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Registered: 2010-04-18
Posts: 2

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)

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#17 2010-04-18 21:04:41

rzaidan
Member
Registered: 2009-08-13
Posts: 59

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

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#18 2010-04-18 21:14:13

rzaidan
Member
Registered: 2009-08-13
Posts: 59

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

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#19 2010-04-19 01:02:18

ganesh
Administrator
Registered: 2005-06-28
Posts: 37,090

Re: Logarithms

Gentleman,

Thanks for the proofs.
Regards.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#20 2010-08-16 16:15:11

jonnyj99
Member
Registered: 2010-08-16
Posts: 4

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}. \,

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#21 2011-05-27 19:59:31

reconsideryouranswer
Member
Registered: 2011-05-11
Posts: 171

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 (2011-05-27 20:05:01)


Signature line:

I wish a had a more interesting signature line.

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#22 2011-05-27 23:57:00

ganesh
Administrator
Registered: 2005-06-28
Posts: 37,090

Re: Logarithms

Hi reconsideryouranswer,

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


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#23 2017-07-02 02:41:41

iamaditya
Member
From: Planet Mars
Registered: 2016-11-15
Posts: 821

Re: Logarithms

Hi all,

I saw this post today and saw the probs on log. Well, they are not bad, they are good. But you can also try these problems here by me (Credit: to a book):
http://www.mathisfunforum.com/viewtopic … 93#p399193


Practice makes a man perfect.
There is no substitute to hard work
All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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#24 2017-09-24 11:06:31

greg1313
Member
Registered: 2016-12-19
Posts: 20

Re: Logarithms

JaneFairfax wrote:

JaneFairfax, here is a basic proof of L4:

For all real a > 1, y = a^x is a strictly increasing function.

log(base 2)3 versus log(base 3)5

2*log(base 2)3 versus 2*log(base 3)5

log(base 2)9 versus log(base 3)25

2^3 = 8 < 9

2^(> 3) = 9

3^3 = 27 > 25

3^(< 3) = 25

So, the left-hand side is greater than the right-hand side, because
Its logarithm is a larger number.

Last edited by greg1313 (2019-10-30 07:44:37)

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