document.write( "Question 371352: Use the logarithm properties to condense:\r
\n" ); document.write( "\n" ); document.write( "1/3[2 ln(x+5) - lnx - ln(x^2-4)] \n" ); document.write( "
Algebra.Com's Answer #264871 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"%281%2F3%29%282%2Aln%28x%2B5%29+-+ln%28x%29+-+ln%28x%5E2-4%29%29\"
\n" ); document.write( "The three logarithms in the parentheses are not like terms so we cannot subtract them. However there is a property of logarithms, \"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\", which allows us to combine two logarithms into one if all of the following are true:
  • There is a mimus between them.
  • The bases of the logarithms are the same.
  • The coefficient of each logarithm is a 1.

\n" ); document.write( "Your logarithms meet the first two requirements. But the first logarithm has a coefficient of 2. Fortunately there is another property of logarithms, \"q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29\", which allows us to \"move\" a coefficient into the argument as an exponent. So we start by using this property on the first logarithm:
\n" ); document.write( "\"%281%2F3%29%28ln%28%28x%2B5%29%5E2%29+-+ln%28x%29+-+ln%28x%5E2-4%29%29\"
\n" ); document.write( "Now we can use the first property on the first two logarithms:
\n" ); document.write( "\"%281%2F3%29%28ln%28%28x%2B5%29%5E2%2Fx%29+-+ln%28x%5E2-4%29%29\"
\n" ); document.write( "Next we can use the first property again to combine the remaining logarithms:
\n" ); document.write( "\"%281%2F3%29%28ln%28%28%28x%2B5%29%5E2%2Fx%29%2F%28x%5E2-4%29%29%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"%281%2F3%29%28ln%28%28%28x%2B5%29%5E2%29%2F%28x%2A%28x%5E2-4%29%29%29%29\"
\n" ); document.write( "This may be condensed enough. But we can use the second property again to move the coefficient of 1/3:
\n" ); document.write( "\"ln%28%28%28%28x%2B5%29%5E2%29%2F%28x%2A%28x%5E2-4%29%29%29%5E%281%2F3%29%29\"
\n" ); document.write( "This may be the desired answer. Alternatively, since 1/3 as an exponent means \"cube root\", we could rewrite this as:
\n" ); document.write( "\"ln%28root%283%2C+%28%28x%2B5%29%5E2%29%2F%28x%2A%28x%5E2-4%29%29%29%29\" \n" ); document.write( "
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