SOLUTION: i really don't know where to begin with this problem at all? given log base a 2= 0.431 and log base a 3= 0.683 find the following .... log base a 6 and log base a 81 and log bas

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: i really don't know where to begin with this problem at all? given log base a 2= 0.431 and log base a 3= 0.683 find the following .... log base a 6 and log base a 81 and log bas      Log On


   

Question 115571: i really don't know where to begin with this problem at all?
given log base a 2= 0.431 and log base a 3= 0.683 find the following ....
log base a 6 and log base a 81 and log base a 2a?

if you could help me it would be great thank you !
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
You need to know a couple of properties of logarithms.
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One of the properties is the multiplication property: log%28a%2C+x%2Ay%29+=+log%28a%2Cx%29+%2B+log%28a%2Cy%29
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The first problem is to find log%28a%2C6%29. Note that 6 is equal to 2*3. So you can re-write
the problem as log%28a%2C2%2A3%29 But the property of logs described above converts this to:
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log%28a%2C2%29+%2B+log%28a%2C3%29
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Both of these logs are given ... log%28a%2C2%29+=+0.341 and log%28a%2C3%29+=+0.683 Inserting
these values into the problem results in:
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log%28a%2C2%2A3%29=log%28a%2C2%29+%2B+log%28a%2C3%29=+0.341+%2B+0.683+=+1.024
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So the answer is log%28a%2C6%29+=+1.024
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The next problem uses the exponent property that log%28a%2Cm%5En%29+=+n%2Alog%28a%2Cm%29
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The problem is to find the value of log%28a%2C81%29. To do that you can recognize that
81+=+3%5E4. If you substitute that into the problem, you get:
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log%28a%2C81%29+=+log%28a%2C3%5E4%29
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Next apply the exponent property to add another step to the problem:
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log%28a%2C81%29+=+log%28a%2C3%5E4%29+=+4%2Alog%28a%2C3%29
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Finally add the last step by substituting 0.683 for log%28a%2C3%29:
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log%28a%2C81%29+=+log%28a%2C3%5E4%29+=+4%2Alog%28a%2C3%29=+4%2A0.683+=+2.732
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So the answer to this second problem is log%28a%2C81%29+=+2.732
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The final problem is:
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log%28a%2C2%2Aa%29
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Use the multiplication property to split this into two logarithms as follows:
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log%28a%2C2%2Aa%29=log%28a%2C2%29%2Blog%28a%2Ca%29
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Another property is that for any base if you take the logarithm of the base, the answer is 1.
This means log%28a%2Ca%29+=+1. Plus you are given that log%28a%2C2%29+=+0.431. Making these
two substitutions results in:
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log%28a%2C2%2Aa%29=log%28a%2C2%29%2Blog%28a%2Ca%29=0.431+%2B+1+=+1.431.
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So the answer to this problem is: log%28a%2C2%2Aa%29+=+1.431
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Hope this helps you to gain some insight into a couple of properties of logarithms. It's
sort of late to be doing math, so be sure to check the math in these problems. Mistakes
have a tendency to creep into late-night work ... but the basic principles involving the
properties of logarithms are correct.
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