SOLUTION: log base x+2 (4-x)= 1+ log base x+2 (x)

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Question 1128869: log base x+2 (4-x)= 1+ log base x+2 (x)
Found 2 solutions by greenestamps, Alan3354:
Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


log%28x%2B2%2C%284-x%29%29+=+1%2B+log%28x%2B2%2C%28x%29%29

Get all the terms of the equation in terms of log base x+2:

log%28x%2B2%2C%284-x%29%29=+log%28x%2B2%2C%28x%2B2%29%29+%2B+log%28x%2B2%2C%28x%29%29

Use the rule for addition of logs:

log%28x%2B2%2C%284-x%29%29=+log%28x%2B2%2C%28x%28x%2B2%29%29%29

Equate the arguments and solve:

4-x+=+x%28x%2B2%29
4-x+=+x%5E2%2B2x
x%5E2%2B3x-4+=+0
%28x%2B4%29%28x-1%29+=+0
x+=+-4 or x+=+1

The base of the logarithm is either -4+2 = -2 or 1+2 = 3. -2 doesn't work for the base of a logarithm; so the base of the logarithm is 3.

ANSWER: x = 1; the base of the logarithm is 3.

CHECK:
log%28x%2B2%2C%284-x%29%29+=+log%283%2C%283%29%29+=+1
1%2B+log%28x%2B2%2C%28x%29%29+=+1%2Blog%283%2C%281%29%29+=+1%2B0+=+1

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
log base x+2 (4-x)= 1+ log base x+2 (x)
Assuming x+2 is the base:
-------------
log%28%28x%2B2%29%2C%284-x%29%29+=+1%2B+log%28%28x%2B2%29%2Cx%29

log%28%28x%2B2%29%2C%284-x%29%29+=+log%28%28x%2B2%29%2Cx%2A%28x%2B2%29%29
4-x+=+x%2A%28x%2B2%29+=+x%5E2%2B2x
x%5E2+%2B+3x+-+4+=+0
(x+4)*(x-1) = 0
x = -4 ------------ Not allowed
x = 1