SOLUTION: How do you simplify 4In2 + 1/2 In4 - In8 to a single logarithm?

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Question 70927: How do you simplify 4In2 + 1/2 In4 - In8 to a single logarithm?
Found 2 solutions by algebrapro18, bucky:
Answer by algebrapro18(249) About Me  (Show Source):
You can put this solution on YOUR website!
4In2 + 1/2 In4 - In8
ln(2^4)+ln(4^(1/2))-ln(8)
ln(16)+ln(2)-ln(8)
ln([16*2]/8)
ln 4

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
4%2Aln2+%2B+%281%2F2%29%2Aln4-ln8
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Become familiar with three rules of logarithms:
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(1) The log of a product can be written as the sum of the two logarithms of the multiplicand and the multiplier. And the reverse is true. If you have the sum of two logarithms they can be changed to the logarithm of the product of the two terms.
Examples: log(6) can be written as log(3*2) which can be expanded to log(3) + log(2).
And going in reverse log(5) + log(3) = log(5*3) = log(15)
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(2) The log of a division can be written as the difference of the two logarithms of the dividend and the divisor. And the reverse is true. If you have the difference of two logarithms they can be changed to the logarithm of the division of the two terms.
Examples: log(3) can be written as log(6/2) which can be expanded to log(6) - log(2).
And going in reverse log(12) - log(4) = log(12/4) = log(3)
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(3) The exponent of a term whose log is to be found can be moved so that it becomes the multiplier of the logarithm. And in reverse, the multiplier of a logarithm can be removed as the multiplier and used as the exponent of the term.
Examples: log(3^2) is the same as 2*log(3). And in reverse, 3*log(5) is the same as log(5^3).
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We will use these rules to simplify your problem. You were given:
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4%2Aln%282%29+%2B+%281%2F2%29%2Aln%284%29-ln%288%29
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Notice in the middle term the ln(4) can be written as the ln(2*2) which breaks out into ln(2) + ln(2) by using rule 1 above. These two logs are identical and can therefore be added to 2*ln(2). Substitute this into the equation to get:
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4%2Aln%282%29+%2B+%281%2F2%29%2A%282%2Aln%282%29%29-ln%288%29
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Now in the middle term multiply the (1/2) times the 2 to get 1. This makes the equation become:
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4%2Aln%282%29+%2B+ln%282%29+-+ln%288%29
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Observe that the first two terms can be added to result in 5*ln(2) and this reduces the problem to:
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5%2Aln2+-+ln%288%29
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In the second term you can recognize that 8 is equal to 2%5E3. Therefore, replace the 8 with 2%5E3 and the equation becomes:
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5%2Aln%282%29+-+ln%282%5E3%29
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But using rule 3 above we can take the exponent 3 away and make it the multiplier of the ln. When we do, the equation becomes:
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5%2Aln%282%29+-+3%2Aln%282%29
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Since the ln terms are the same, the two terms can be directly subtracted to result in:
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5%2Aln%282%29+-+3%2Aln%282%29+=+%285-3%29%2Aln%282%29+=+2%2Aln%282%29
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So the answer to your problem is that the entire expression you were given can be reduced to 2%2Aln%282%29
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I hope this helps you to see how to simplify logarithm problems by using the three rules above.