shows up twice. (√(5) - 1)/2) and 1.61803]]>

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^....?

]]>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.

]]>can someone explain this to me?

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