document.write( "Question 114927: Please help solve this...\r
\n" ); document.write( "\n" ); document.write( "log9(x-7)+log9(x-7)=1 \n" ); document.write( "
Algebra.Com's Answer #83623 by bucky(2189)\"\" \"About 
You can put this solution on YOUR website!
Given:
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\n" ); document.write( "\"log%289%2C%28x-7%29%29+%2B+log%289%2C%28x-7%29%29=1\"
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\n" ); document.write( "Since the base of the two logarithms on the left side is the same, the fact that they are
\n" ); document.write( "added means that by the rules of logarithms they can be combined to a single logarithm of
\n" ); document.write( "their products. In other words they combine as follows:
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\n" ); document.write( "You can get the same result if you look at it this way ...
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\n" ); document.write( "Recognize that \"log%289%2C%28x-7%29%29+%2B+log%289%2C%28x-7%29%29\" involves two identical terms and so the sum
\n" ); document.write( "just equals 2 times one of the terms. In other words the sum is \"2%2Alog%289%2C%28x-7%29%29\".
\n" ); document.write( "And by the rules of logarithms, the multiplier of a logarithm can be used as an exponent of
\n" ); document.write( "the quantity on which the logarithm is operating. So the multiplier 2 becomes the exponent
\n" ); document.write( "2 and the sum of the two logarithms equals \"2%2Alog%289%2C%28x-7%29%29\" which in turn equals
\n" ); document.write( "\"log%289%2C%28x-7%29%5E2%29\"
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\n" ); document.write( "Back to the problem. We have now found an equivalent form of the left side of the equation
\n" ); document.write( "and therefore the equation becomes:
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\n" ); document.write( "\"log%289%2C%28x-7%29%5E2%29=+1\"
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\n" ); document.write( "Now convert from this logarithmic form to the exponential form of this equation using the
\n" ); document.write( "conversion rule:
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\n" ); document.write( "\"log%28a%2CB%29=+y\" is equivalent to the exponential form \"a%5Ey+=+B\"
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\n" ); document.write( "In our problem: a = 9, B = (x - 7)^2, and y = 1. Substituting these values into the exponential
\n" ); document.write( "form of the logarithmic equation results in our equation becoming:
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\n" ); document.write( "\"9%5E1+=+%28x-7%29%5E2\"
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\n" ); document.write( "Raising 9 to the exponent 1 just makes it 9 so the equation simplifies to:
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\n" ); document.write( "\"9+=+%28x+-+7%29%5E2\"
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\n" ); document.write( "Take the square root of both sides. The square root of 9 is 3 and the square root of
\n" ); document.write( "\"%28x-7%29%5E2+=+x-7\". Substituting this information into the equation reduces the equation to:
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\n" ); document.write( "\"3+=+x+-+7\"
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\n" ); document.write( "Solve for x by getting rid of the -7 on the right side. To do that just add +7 to both
\n" ); document.write( "sides and you have:
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\n" ); document.write( "\"10+=+x\"
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\n" ); document.write( "Hope this helps you see your way through this problem.
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