SOLUTION: 2 log(x+2)+log4 = log x + 4 log3

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Question 33989: 2 log(x+2)+log4 = log x + 4 log3
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
i assume that the logs are all base10? If so,

2log(x+2)+log4 = log x + 4 log3
2log(x+2) = log x + 4 log3 - log4
2log(x+2) - log x = 4 log3 - log4
+log%28x%2B2%29%5E2+-+log+x+=+log3%5E4+-+log4+
+log%28%28x%2B2%29%5E2%2Fx%29+=+log%283%5E4%2F4%29+

Now, raise both terms to power 10, to cancel the logs:
+%28x%2B2%29%5E2%2Fx+=+3%5E4%2F4+
+%28x%2B2%29%5E2%2Fx+=+81%2F4+
+4%28x%2B2%29%5E2+=+81x+
+4%28x%5E2%2B4x%2B4%29+=+81x+
+4x%5E2%2B16x%2B16+=+81x+
+4x%5E2-65x%2B16+=+0+

This is just a simple quadratic now, to either factorise if you can spot it or use the quadratic formula if not.

+x+=+%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29+
+x+=+%28-%28-65%29+%2B-+sqrt%28%28-65%29%5E2-4%284%29%2816%29%29%29%2F%282%284%29%29+

Working this through gives x=16 or x=0.25

jon.