Math Is Fun Forum

MajikWaffle

Member

Registered: 2005-12-14

Posts: 11

Log Question

http://img228.imageshack.us/img228/1799/untitled8av.png

can someone explain this to me?

ryos

Member

Registered: 2005-08-04

Posts: 394

Re: Log Question

First rewrite them in exponential form:

9^x = a
12^x = b
16^x = a + b  →  16^x = 9^x + 12^x
Solve for x.

You know, I'm not sure that's even possible without the aid of technology. My calculator says that:
x = ln[ (√(5) - 1)/2) ] / ln(3/4).
That's log base 3/4 of (√(5) - 1)/2), which is 1.67272.

Guess what? 12^1.67272 / 9^1.67272  =  63851199804262/39462211701495.  More sanely, it = 1.61803.

Anyone who can solve 16^x = 9^x + 12^x for x by hand deserves a trophy.

Last edited by ryos (2005-12-14 19:22:31)

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El que pega primero pega dos veces.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Log Question

Yea, that is quite an equation, I don't believe I've ever had to solve one like it before.

My calculator comes up with 9^x = 16^(ln(3)x / 2*ln(2)) and 12^x = 16^(ln(12)x/4*ln(2))

This makes the equation:

16^x = 16^(ln(3)x / 2*ln(2)) + 16^(ln(12)x/4*ln(2))

That seems to be a step in the right direction, getting a common base.  And can anyone derive 9^x = 16^... and 12^x = 16^....?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Log Question

would not b / a = (4 / 3)^x be a suitable answer?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Log Question

I don't know why, but in post #2 by ryos, the golden ratio
shows up twice.   (√(5) - 1)/2)  and 1.61803

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igloo myrtilles fourmis

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Log Question

Maybe he just has the midas touch.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."