document.write( "Question 421653: Express loga36 + ½ loga256 - 2 loga48 as a single logarithm
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Algebra.Com's Answer #294422 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"log%28a%2C+%2836%29%29+%2B+%281%2F2%29log%28a%2C+%28256%29%29+-+2log%28a%2C+%2848%29%29\"
\n" ); document.write( "These are not like terms so we cannot add and subtract to combine them. (Like logarithmic terms have the same bases and the same arguments. Your logarithms all have the same base, a, but they have different arguments, 36, 256 and 48.)

\n" ); document.write( "However there are some properties of logarithms:
  • \"log%28z%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28z%2C+%28p%2Aq%29%29\"
  • \"log%28z%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28z%2C+%28p%2Fq%29%29\"
which provide another way to combine logarithms. These properties require that the bases be the same and the coefficients be 1's. Your logarithms have the same base but the 2nd and 3rd logarithms have coefficients of 1/2 and 2, respectively. Fortunately there is yet another property of logarithms, \"q%2Alog%28z%2C+%28p%29%29+=+log%28z%2C+%28p%5Eq%29%29\", which provides a way to move a coefficient of a logarithm into the argument as its exponent. So we will use this third property to \"move\" the coefficients of the last two logs out of the way:
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\n" ); document.write( "Since 1/2 as an exponent means square root and since \"256+=+16%5E2\", the middle argument simplifies to 16:
\n" ); document.write( "\"log%28a%2C+%2836%29%29+%2B+log%28a%2C+%2816%29%29+-+log%28a%2C+%2848%5E2%29%29\"
\n" ); document.write( "Since the coefficients are all 1's, we can now start using the first two properties to combine them. We use the first property on the first two logs because they have a \"+\" between the logs:
\n" ); document.write( "\"log%28a%2C+%2836%2A16%29%29+-+log%28a%2C+%2848%5E2%29%29\"
\n" ); document.write( "Now we can use the second property to combine the remaining logs because they both have a \"-\" between the logs:
\n" ); document.write( "\"log%28a%2C+%28%2836%2A16%29%2F48%5E2%29%29\"
\n" ); document.write( "The only thing left is to reduce the fraction:
\n" ); document.write( "\"log%28a%2C+%28%283%2A3%2A4%2A16%29%2F%2816%2A3%2A4%2A4%2A3%29%29%29\"
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\n" ); document.write( "\"log%28a%2C+%281%2F4%29%29\" \n" ); document.write( "
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