SOLUTION: 5^x - 3^x = 16, find x.

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Question 1209381: 5^x - 3^x = 16,
find x.
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!

We can find an answer x=2, by trial and error,

5%5E2-3%5E2=25-9=16

But is that the only solution, and if so, why?

5%5Ex+-+3%5Ex+=+16

5%5Ex%2F5%5Ex-3%5Ex%2F5%5Ex=16%2F5%5Ex

1-%283%2F5%29%5Ex=16%2F5%5Ex

1=16%2F5%5Ex%2B%283%2F5%29%5Ex

The function y=16%5E%22%22%2F5%5Ex%2B%283%2F5%29%5Ex
is strictly decreasing.  It has value 17 when x=0, 17 > 1, 
and value 43/125=0.344 when x=3, 0.344 < 1

Thus it can intersect the horizontal line y=1 only once.

So it has only one solution, x=2.

Edwin