Question 115571
You need to know a couple of properties of logarithms.
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One of the properties is the multiplication property: {{{log(a, x*y) = log(a,x) + log(a,y)}}}
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The first problem is to find {{{log(a,6)}}}. Note that 6 is equal to 2*3. So you can re-write 
the problem as {{{log(a,2*3)}}} But the property of logs described above converts this to:
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 {{{log(a,2) + log(a,3)}}}
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Both of these logs are given ... {{{log(a,2) = 0.341}}} and {{{log(a,3) = 0.683}}} Inserting
these values into the problem results in:
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{{{log(a,2*3)=log(a,2) + log(a,3)= 0.341 + 0.683 = 1.024}}}
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So the answer is {{{log(a,6) = 1.024}}}
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The next problem uses the exponent property that {{{log(a,m^n) = n*log(a,m)}}}
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The problem is to find the value of {{{log(a,81)}}}. To do that you can recognize that
{{{81 = 3^4}}}. If you substitute that into the problem, you get:
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{{{log(a,81) = log(a,3^4)}}}
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Next apply the exponent property to add another step to the problem:
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{{{log(a,81) = log(a,3^4) = 4*log(a,3)}}}
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Finally add the last step by substituting 0.683 for {{{log(a,3)}}}:
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{{{log(a,81) = log(a,3^4) = 4*log(a,3)= 4*0.683 = 2.732}}}
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So the answer to this second problem is {{{log(a,81) = 2.732}}}
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The final problem is:
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{{{log(a,2*a)}}}
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Use the multiplication property to split this into two logarithms as follows:
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{{{log(a,2*a)=log(a,2)+log(a,a)}}}
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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(a,a) = 1}}}. Plus you are given that {{{log(a,2) = 0.431}}}. Making these
two substitutions results in:
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{{{log(a,2*a)=log(a,2)+log(a,a)=0.431 + 1 = 1.431}}}. 
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So the answer to this problem is: {{{log(a,2*a) = 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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