document.write( "Question 365204: Write the expression as a single logarithm. Express powers as factors. (Simplify your answer)
\n" ); document.write( "ln(x/(x-2))+ln((x+2)/x)-ln((x^2)-4) \n" ); document.write( "
Algebra.Com's Answer #260487 by jsmallt9(3758)\"\" \"About 
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I assume the expression is:
\n" ); document.write( "\"ln%28x%2F%28x-2%29%29%2Bln%28%28x%2B2%29%2Fx%29-ln%28x%5E2-4%29\"
\n" ); document.write( "These are not like terms because the arguments are different. But there are properties of logarithms which allow us to combine these logarithms into one:
  • \"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" ); document.write( "With these properties all you need is the same base and coefficients of 1 and your logarithms meet both requirements. Using the first property on the first two logarithms (because of the \"+\" between them) we get:
\n" ); document.write( "\"ln%28x%2F%28x-2%29%29%2A%28%28x%2B2%29%2Fx%29-ln%28x%5E2-4%29\"
\n" ); document.write( "The x's cancel:
\n" ); document.write( "\"ln%28cross%28x%29%2F%28x-2%29%29%2A%28%28x%2B2%29%2Fcross%28x%29%29-ln%28x%5E2-4%29\"
\n" ); document.write( "leaving:
\n" ); document.write( "\"ln%28%28x%2B2%29%2F%28x-2%29%29-ln%28x%5E2-4%29\"
\n" ); document.write( "Now we can use the second property (because of the \"-\" between them):
\n" ); document.write( "\"ln%28%28%28x%2B2%29%2F%28x-2%29%29%2F%28x%5E2-4%29%29\"
\n" ); document.write( "Now we try to simplify the compllex fraction. We start by factoring the denominator:
\n" ); document.write( "\"ln%28%28%28x%2B2%29%2F%28x-2%29%29%2F%28%28x%2B2%29%28x-2%29%29%29\"
\n" ); document.write( "Now we will multiply the numerator and denominator by (x-2):
\n" ); document.write( "
\n" ); document.write( "The (x-2)'s in the numerator cancel:
\n" ); document.write( "
\n" ); document.write( "leaving:
\n" ); document.write( "\"ln%28%28%28%28x%2B2%29%2F1%29%2F%28%28x%2B2%29%28x-2%29%29%29%281%2F%28x-2%29%29%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"ln%28%28x%2B2%29%2F%28%28x%2B2%29%28x-2%29%28x-2%29%29%29\"
\n" ); document.write( "Now the (x+2)'s cancel:
\n" ); document.write( "\"ln%28cross%28%28x%2B2%29%29%2F%28cross%28%28x%2B2%29%29%28x-2%29%28x-2%29%29%29\"
\n" ); document.write( "leaving:
\n" ); document.write( "\"ln%281%2F%28%28x-2%29%28x-2%29%29%29\"
\n" ); document.write( "Since it says not to use powers, we will not change the denominator to \"%28x-2%29%5E2\". This may be an acceptable answer. However we can simplify a little further. We can rewrite the argument as follows:
\n" ); document.write( "\"ln%28%28%28x-2%29%28x-2%29%29%5E%28-1%29%29\"
\n" ); document.write( "and then use the thrid property of logarithms,
\n" ); document.write( "or
\n" ); document.write( "\"-ln%28%28x-2%29%28x-2%29%29%29\"
\n" ); document.write( " \n" ); document.write( "
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