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, the negative log becomes the denominator and the positive log the numerator.
Applying this rule leads to the given equation becoming:
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Next convert this to exponential form by raising the base (which is 10 for this problem) to
the power of the term on the right side of the equation (which is 1) and setting it equal
to the quantity that the log operator is acting on. In other words:
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And since the equation is:
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Get rid of the denominator by multiplying both sides of the equation by the quantity (x + 1)
to change the equation to:
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Multiplying out the right side results in the equation becoming:
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Get rid of the 10x on the right side by subtracting 10x from both sides:
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Next get rid of the 34 on the left side by subtracting 34 from both sides:
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Solve for x by dividing both sides of this equation by -8 ... which is the multiplier
of x to get:
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So the answer is x = 3
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Check by returning to the original problem and substituting +3 for x to see if the two
log terms combine to equal 1:
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Start with:
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Substitute +3 for x and it becomes:
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In the first log term multiply 2 times 3 and add the 34 to that product to get 40. This reduces
the equation to:
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In the second log term add the 3 and 1 to make the equation:
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Calculator time!!! On a scientific calculator find the log to the base 10 of 40 by entering
40 and pressing the "log" key. You should get 1.602059991
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Next find the log to the base 10 of 4 by entering 4 and pressing the "log" key to get
0.602059991.
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Subtract the two logs:
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1.602059991 - 0.602059991
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And you do, indeed, find that the difference is 1, just as the problem said it should be.
Therefore, the answer of x = 3 does work out, and that is the correct answer.
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Hope this helps you to understand the problem and the processes you can use to solve it.
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