SOLUTION: Please explain how to work this problem: log of(x+1), base 2 plus 2 times the log of(x-1), base 4 equals 3

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Please explain how to work this problem: log of(x+1), base 2 plus 2 times the log of(x-1), base 4 equals 3      Log On


   

Question 173045: Please explain how to work this problem:
log of(x+1), base 2 plus 2 times the log of(x-1), base 4 equals 3
Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
i think i have a solution for you.
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before i go any further, your answer is:
x = 3
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i checked that out in the original formulas and it works.
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the problem:
log%282%2C%28x%2B1%29%29+%2B+2%2Alog%284%2C%28x-1%29%29+=+3
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so i could see the problem a little easier, i let:
a = (x+1)
and
b = (x-1)
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the problem then looks like:
log%282%2Ca%29+%2B+2%2Alog%284%2Cb%29+=+3
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i then wanted to get a common base, so i converted the logarithm with a base of 4 to a logarithm with a base of 2 using the conversion formula:
log%284%2Cb%29+=+log%282%2Cb%29%2Flog%282%2C4%29
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the problem then looks like:
log%282%2Ca%29+%2B+2+%2A+%28log%282%2Cb%29%2Flog%282%2C4%29%29+=+3
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since log%282%2C4%29+=+d if and only if 2%5Ed+=+4 and i can show that d has to be equal to 2, then:
log%282%2C4%29+=+2
and the problem now looks like:
log%282%2Ca%29+%2B+2%2A%28log%282%2Cb%29%2F2%29+=+3
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this simplifies to:
log%282%2Ca%29+%2B+log%282%2Cb%29+=+3
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since the laws of logarithms state:
log%282%2Ca%2Ab%29+=+log%282%2Ca%29+%2B+log%282%2Cb%29
then:
log%282%2Ca%29+%2B+log%282%2Cb%29 must equal log%282%2Ca%2Ab%29
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the problem now looks like:
log%282%2Ca%2Ab%29+=+3
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the basic laws of logarithms state:
log%282%2Ca%2Ab%29+=+3 if and only if:
2%5E3+=+a%2Ab
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since 2%5E3 = 8, the problem now looks like:
a%2Ab+=+8
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if you recall from way back, i let:
a = (x+1)
b = (x-1)
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i can now substitute (x+1) for a, and (x-1) for b, and the problem now looks like:
%28x%2B1%29%2A%28x-1%29+=+8
multiplying out the factors, this becomes:
x%5E2+-1+=+8
add 1 to both sides of the equation to get:
x%5E2+=+9
take the square root of both sides of the equation to get:
x+=+3
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that's your answer.
it was not a simple problem.
you needed to be able to convert bases: log%284%2Cb%29+=+log%282%2Cb%29%2Flog%282%2C4%29
you also needed to remember: log%282%2Ca%29+%2B+log%282%2Cb%29+=+log%282%2Ca%2Ab%29
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fyi:
you could have solved log%282%2C4%29 by converting to log%2810%2C4%29%2Flog%2810%2C2%29 as well as the way i showed you above. then you could have used the log function of the calculator to get your answer.
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