Answer: D) x = 4
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Explanation:
We'll use these two log rules
- log(A)+log(B) = log(A*B)
- If log(A) = log(B), then A = B
So,
log(2x) + log(x-1) = log(6x)
log[ 2x*(x-1) ] = log(6x) ...... use rule 1
2x*(x-1) = 6x ...... use rule 2
2x^2-2x = 6x
2x^2-2x-6x = 0
2x^2-8x = 0
2x(x-4) = 0
2x = 0 or x-4 = 0
x = 0 or x = 4
The two possible solutions are x = 0 or x = 4. However, we must check both possible answers because some or all of them may be extraneous.
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Checking x = 0
log(2x) + log(x-1) = log(6x)
log(2*0) + log(0-1) = log(6*0)
log(0) + log(-1) = log(0)
We run into a roadblock. We cannot take the log of 0 or anything negative. The domain of y = log(x) is x > 0. So we cross x = 0 off the list.
Now let's check x = 4
log(2x) + log(x-1) = log(6x)
log(2*4) + log(4-1) = log(6*4)
log(8) + log(3) = log(24)
log(8*3) = log(24)
log(24) = log(24)
We've confirmed x = 4. This is the only solution.
