SOLUTION: find the two values of x that satisfy the equation: 2log(base4)X-log(base4)(3X-4))=1/2

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Question 743558: find the two values of x that satisfy the equation:
2log(base4)X-log(base4)(3X-4))=1/2
Found 2 solutions by nerdybill, josmiceli:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
2log(base4)X-log(base4)(3X-4)=1/2
log(base4)X^2-log(base4)(3X-4)=1/2
log(base4)X^2/(3X-4)=1/2
X^2/(3X-4)=4^(1/2)
X^2/(3X-4)=2
X^2=2(3X-4)
X^2=6X-8
X^2-6X+8=0
(X-2)(X-4)=0
X = {2, 4}

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+2%2Alog%284%2Cx%29+-+log%284%2C+3x+-+4%29+=+1%2F2+
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The key is to make the substitution
+1%2F2+=+log%28+4%2C2+%29+
+log%284%2Cx%5E2%29+-+log%284%2C+3x+-+4%29+=+log%28+4%2C2+%29+
+log%28+4%2C+%28x%5E2+%2F+%28+3x+-+4%29%29+%29+=+log%28+4%2C2+%29+
+x%5E2+%2F+%28+3x-4+%29++=+2+
+x%5E2+=+2%2A%28+3x-4+%29+
+x%5E2+=+6x+-+8+
+x%5E2+-+6x+%2B+8+=+0+
+%28+x+-+4+%29%2A%28+x+-+2+%29+=+0+ ( by inspection )
+x+=+4+
+x+=+2+
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Check the answers:
+2%2Alog%284%2Cx%29+-+log%284%2C+3x+-+4%29+=+1%2F2+
+2%2Alog%284%2C4%29+-+log%284%2C+3%2A4+-+4%29+=+1%2F2+
+2%2A1+-+log%28+4%2C+8+%29+=+1%2F2+
+log%28+4%2C8+%29+=+2+-+1%2F2+
+log%28+4%2C8+%29+=+3%2F2+
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This is true since
+4%5E%283%2F2%29+=+8+
Square both sides
+4%5E3+=+64+
+64+=+64+
OK
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