# Math Is Fun Forum

Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,703

### 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
Real Member
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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,703

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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,703

### 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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,703

### 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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,703

### 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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,703

### Re: Logarithms

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

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

L # 4

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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,703

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

### Re: Logarithms

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

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

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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,703

### Re: Logarithms

No, I didn't
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!!

Last edited by JaneFairfax (2009-01-06 00:30:04)

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

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

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

keveenjones
Member
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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,703

### 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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,703

### 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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