You can put this solution on YOUR website! I presume that you are working with logarithms having the base 10.
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You are asked to solve for x in the equation:
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Collect the logarithmic terms on the left side of the equation by subtracting from both sides as follows:
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On the right side the two logarithmic terms are of opposite signs and therefore the cancel each other out. When you do that cancelling subtraction, the equation reduces to:
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By the rules of logarithms, the subtraction of two logarithms is equivalent to the logarithm of their quotient (in which the quantity in the subtracted logarithm becomes the divisor or denominator). This means that the left side of the preceding equation is equivalent to the left side shown below, and the right side of the equation remains as 1.
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Now convert this to the exponential form of a logarithm. Do that by raising the base (in this case 10) to the power of the right side of this logarithmic equation (in this problem that is the 1 on the right side) and set that result equal to the quantity the logarithm is operating on ( in this problem that is the quotient of (7x+1)/(x-2)). In equation form this becomes:
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Just to make this be a little more conventional in form, transpose this equation (switch the sides) to get:
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The right side has 10 raised to the first power, and that is just 10. So you can eliminate the exponent 1 and just write:
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Then get rid of the denominator on the left side by multiplying both sides of this equation by (x - 2) as follows:
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Cancel the common term in the numerator and the denominator on the left side:
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and you are left with:
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Do the distributed multiplication on the right side by multiplying 10 times each of the terms in the parentheses to get:
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Collect the terms having an x on the left side of the equation by subtracting 10x from both sides. Also collect the constants on the right side by subtracting 1 from both sides. This is done as follows:
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Combine the like terms on both sides. On the left side the 7x - 10x results in -3x and the +1 - 1 results in zero and need not be shown on the left side. On the right side the 10x - 10x results in zero and need not be shown and the -20 and -1 combine to -21. So the equation is simplified to:
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Solve for x by dividing both sides of this equation by -3 and you get:
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and this results in the answer:
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Hope this helps you to become a little more familiar with some of the rules you can use to solve logarithmic equations.
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