SOLUTION: how do i solve this: log(3x+8)=1+log(x-1)

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Question 533786: how do i solve this: log(3x+8)=1+log(x-1)
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given to solve for x: log%2810%2C3x%2B8%29=1%2Blog%2810%2Cx-1%29
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First, gather all of the log terms on the left side of the equation by subtracting log%2810%2Cx-1%29 from both sides. When you do that this equation becomes:
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log%2810%2C3x%2B8%29+-+log%2810%2Cx-1%29+=+1
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By the rules of logarithms the difference of two logarithms of differing quantities equals the log of division of the two quantities. In other words the above equation is equal to:
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log%2810%2C%283x%2B8%29%2F%28x-1%29%29=+1
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Convert this logarithmic equation to exponential form by raising the base (which is 10) to the exponent of the quantity on the right side of this equation, and set that equal to the quantity on which the logarithm is operating as follows:
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10%5E1+=+%283x%2B8%29%2F%28x-1%29
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Note 10%5E1+=+10
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10+=+%283x%2B8%29%2F%28x-1%29
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Multiply both sides by x-1 to get rid of the denominator:
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10%28x+-+1%29+=+3x%2B8
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Do the distributed multiplication on the left side:
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10x+-+10+=+3x%2B8
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Subtract 3x from both sides:
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7x+-+10+=+8
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Add 10 to both sides:
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7x+=+18
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Solve for x by dividing both sides by 7 to get:
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x+=+18%2F7
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which, by doing the division on the right side (on a calculator) results in:
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x+=+2.57142857
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Check by substituting this value into the original problem that you were given:
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log%2810%2C3%282.57142857%29%2B8%29=1%2Blog%2810%2C2.57142857-1%29
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On the left side multiply the 3 times x to get:
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log%2810%2C7.714285714%2B8%29=1%2Blog%2810%2C2.57142857-1%29
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Do the addition in the left side logarithm and the subtraction in the logarithm term on the right side to get:
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log%2810%2C15.714285714%29+=+1+%2B+log%2810%2C+1.57142857%29
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Using a calculator, determine the value of the two logs:
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1.196294645=1%2B0.196294645
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Do the addition on the right side:
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1.196294645=1.196294645
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So, this shows that when x = 2.57142857, both sides of the original equation that you were given are equal. So the answer x = 2.57142857 is correct.
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Hope this helps you to understand working with logarithms a little better.
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