SOLUTION: log2x+log2(x-6)=4

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Question 22137: log2x+log2(x-6)=4
Found 2 solutions by longjonsilver, MathTherapy:
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
i assume the 2 is the base of the log. You should explain. I shall leave the 2 out, for ease of writing...

log(x)+log(x-6)=4

To remove log-base 2, we need to raise the left and right hand terms to power 2. We can only do that easily when we have 1 term on both sides, so first we re-write the equation as:

log((x)(x-6))=4

And now we raise to power 2, giving

%28x%29%28x-6%29=4%5E2
x%5E2-6x=16
x%5E2-6x-16+=+0
(x-8)(x+2) = 0

so either x=8 or x=-2

jon.

Answer by MathTherapy(10801) About Me  (Show Source):
You can put this solution on YOUR website!
log2x+log2(x-6)=4
*****************
The solution, (x = 8, or x = - 2) by the other person who responded, is PARTIALLY WRONG!!

log+%282%2C+%28x%29%29+%2B+log+%282%2C+%28x+-+6%29%29+=+4
The SMALLER log argument, x - 6, MUST be > 0. So, x - 6 > 0 ===> x > 6.
We then have: log+%282%2C+%28x%29%29+%2B+log+%282%2C+%28x+-+6%29%29+=+4, with x > 6.
                                         log+%282%2C+x%28x+-+6%29%29+=+4 ----- Applying log+%28b%2C+%28c%29%29+%2B+log+%28b%2C+d%29%29 = log+%28b%2C+%28c%2Ad%29%29
                                                  x%28x+-+6%29+=+2%5E4 --- Converting to EXPONENTIAL form <=== Note that the other person has 2%5E4 i/o 4%5E2,
                                                                                                                                                       but both have the same value, 16
                                                 x%5E2+-+6x+=+16 
                                          x%5E2+-+6x+-+16+=+0
                                       (x - 8)(x + 2) = 0 
                                                      x - 8 = 0     OR    x + 2 = 0 ---- Setting each FACTOR equal to 0
                                                            x = 8     OR           x = - 2

The x-value, 8, is > 6, but - 2 is NOT. This makes - 2 an EXTRANEOUS solution!! So, x = 8 is the only VALID/ACCEPTABLE solution!!