SOLUTION: 289^x - 221^x = 169^x, find x.

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Question 1210273: 289^x - 221^x = 169^x,
find x.
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52810) About Me  (Show Source):
You can put this solution on YOUR website!
.

Go to the link

https://www.google.com/search?q=289%5Ex+-+221%5Ex+%3D+169%5Ex&rlz=1C1CHBF_enUS1071US1071&oq=289%5Ex+-+221%5Ex+%3D+169%5Ex&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIHCAEQIRigATIHCAIQIRigATIHCAMQIRigATIHCAQQIRigATIHCAUQIRigATIHCAYQIRiPAjIHCAcQIRiPAtIBCTMwMDJqMGoxNagCCLACAfEFYQOK3wTcqcbxBWEDit8E3KnG&sourceid=chrome&ie=UTF-8

and read there the solution produced by the Google AI.

It is nice, but completely artificial - like artificial flowers, made of paper.



Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

289%5Ex+-+221%5Ex+=+169%5Ex

%2817%5E2%29%5Ex+-+%2813%2A17%29%5Ex+=+%2813%5E2%29%5Ex

%2817%5Ex%29%5E2+-+%2813%5Ex%29%2A17%5Ex+-+13%5E%282x%29=0

That's a quadratic equation in 17x

17%5Ex=%2813%5Ex+%2B-+sqrt%2813%5E%282x%29%2B4%2A13%5E%282x%29%29%29%2F2%5E%22%22 

17%5Ex=%2813%5Ex+%2B-+sqrt%285%2A13%5E%282x%29%29%29%2F2%5E%22%22

17%5Ex=%2813%5Ex+%2B-+13%5Ex%2Asqrt%285%29%29%2F2%5E%22%22 

17%5Ex=%2813%5Ex%2A%281+%2B-+sqrt%285%29%29%29%2F2%5E%22%22

2%2A17%5Ex=13%5Ex%2A%281+%2B-+sqrt%285%29%29

Take natural logs of both sides.
We'll have to use the +

ln%282%29%2Bln%2817%5Ex%29=ln%2813%5Ex%29%2Bln%281+%2B+sqrt%285%29%29

ln%282%29%2Bx%2Aln%2817%29=x%2Aln%2813%29%2Bln%281+%2B+sqrt%285%29%29

x%2Aln%2817%29-x%2Aln%2813%29=ln%281+%2B+sqrt%285%29%29-ln%282%29

x%2A%28ln%2817%29-ln%2813%29%5E%22%22%29=ln%281+%2B+sqrt%285%29%29-ln%282%29

x=%28ln%281+%2B+sqrt%285%29%29-ln%282%29%29%2F%28ln%2817%29-ln%2813%29%29

That's the exact answer.

It's approximately equal to 1.793799575.

Edwin