You can put this solution on YOUR website! Given:
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By the rules of logarithms, the difference of two logs is equal to the log of their quotient.
In other words:
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By applying this rule to the left side of the given problem, you convert the problem to:
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Notice that this tells you that the log of a quantity on the left side equals the log of a
quantity on the right side. For this to be true, the two quantities must be equal because
their logs are equal. Therefore, you can say that:
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Get rid of the denominator on the left side by multiplying both sides by (x + 2) to get:
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Multiply out the right side and you are left with:
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Subtract x + 20 from both sides and you get a quadratic equation:
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combine the + 2x and the -x and you are left with:
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Transpose (switch sides) this equation to get the more conventional quadratic form of:
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Solve this equation by factoring the left side:
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This equation will be true if either or both of the factors on the left side is zero because
a multiplication by zero on the left side makes the entire left side zero and therefore
equal to the right side.
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What value of x makes a factor equal to zero? Set the factor (x + 5) equal to zero and
when you solve the equation you get x = -5. Then set the factor (x - 4) equal to zero and
solve to get x = +4. So you have two possible answers to this problem ... x = -5 and
x = +4.
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But x cannot be a negative number because the log of a negative number is not allowed. So
log(x) = log(-5) is not allowed in the problem. Therefore, this leaves you with the only
possible solution of x = +4, and that is the answer to this problem.
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Check by returning to the original problem and substituting 4 for x to get:
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which simplifies to:
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Use a calculator to find each of these three logs and substitute those values into the
equation. You will find that the left side of the equation does equal the right side, and
therefore, the answer of x = +4 checks.
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Hope this helps you to see your way through the problem.
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