You can put this solution on YOUR website! Given:
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To solve for x, first recognize that the difference of the logs (to the same base) of two quantities
is equal to the log of the division of the two quantities.
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Using this rule of logarithms means you can convert:
. to
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Substitute this conversion into the equation and it becomes:
.
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Look carefully at this. For the left side to be equal to the right side, the two quantities undergoing
the log operator must be equal. That means:
.
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This is a proportion and you can simplify proportions by cross multiplying ... multiplying
the numerator by the denominator on the other side ... for both numerators.
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When you do that you get:
.
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Multiply out both sides and this equation becomes:
.
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Get rid of the x on the right side by subtracting x from both sides to get:
.
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Then get rid of the 3 on the right side by subtracting 3 from both sides to get:
.
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The left side of this equation factors to:
.
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This equation will be true if either of the factors is zero ... which would cause a
multiplication by zero on the left side and therefore make the left side become zero which equals
the right side. Note that one of the factors will be zero if x = +3 or if x = -1.
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Note however that if x equals -1, several of the terms on which the log function is operating
will become negative, and the log of a negative number is not permitted. Therefore,
x = +3 is the only valid answer.
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Hope this helps you to understand the problem and the steps you can go through to solve
for x.
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