can someone explain this to me?
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)
El que pega primero pega dos veces.
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^....?
"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..."
would not b / a = (4 / 3)^x be a suitable answer?
I don't know why, but in post #2 by ryos, the golden ratio
shows up twice. (√(5) - 1)/2) and 1.61803
igloo myrtilles fourmis