document.write( "Question 823741: 7 log6(x−4)+2log6(x+3)−1/2log6 x\r
\n" ); document.write( "\n" ); document.write( "write as the loarithm of a single expression \n" ); document.write( "
Algebra.Com's Answer #495779 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"7log%286%2C+%28x-4%29%29%2B2log%286%2C+%28x%2B3%29%29-%281%2F2%29log%286%2C+%28x%29%29\"
\n" ); document.write( "If these were like terms then we could just add or subtract them to combine them into a single logarithm. (Like logarithmic terms have the same bases and the same arguments.) These, however, are not like terms (same bases, yes, but all three arguments are different: x-4, x+3 and x.)

\n" ); document.write( "Fortunately there are three properties of logarithms which provide an laternate way to combine logarithmic terms:
  • \"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\"
  • \"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\"
  • \"n%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5En%29%29\"
The first two can be used to combine logarithmic terms. (The first one for when the terms have a \"+: between them and the second for when there is a \"-\".) They require that the logs have the same bases and that they have a coefficient of 1.

\n" ); document.write( "Our terms all have coefficients that are not 1's. But that is where the third property comes in. It can be used to \"move\" a coefficient into the argument as its exponent. So that is where we will start: Using the third property to move the coefficients out of the way:
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\n" ); document.write( "Since 1/2 as an exponent is the same as a square root and since fractional exponents do not always display well on algebra.com, I am going to replace the fractional exponent with a square root before proceeding. (This is not something you need to do.)
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\n" ); document.write( "Now we will use the first property to combine the first two terms:
\n" ); document.write( "\"log%286%2C+%28%28x-4%29%5E7+%2A+%28x%2B3%29%5E2%29%29-log%286%2C+%28sqrt%28x%29%29%29\"
\n" ); document.write( "And now we use the second property to combine the remaining terms:
\n" ); document.write( "\"log%286%2C+%28%28%28x-4%29%5E7+%2A+%28x%2B3%29%5E2%29%2Fsqrt%28x%29%29%29\"
\n" ); document.write( "This may be an acceptable answer. But we may want to rationalize the denominator:
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\n" ); document.write( "\"log%286%2C+%28%28%28x-4%29%5E7+%2A+%28x%2B3%29%5E2+%2A+sqrt%28x%29%29%2Fx%29%29\" \n" ); document.write( "
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