document.write( "Question 718443: How do you write 5log[base2]x+(1/2)log[base2]y as a single logarithm \n" ); document.write( "
Algebra.Com's Answer #440974 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"5log%282%2C+%28x%29%29%2B%281%2F2%29log%282%2C+%28y%29%29\"
\n" ); document.write( "We have two logs which we want to write as one log. There are two ways to combine two logarithmic terms into one:
  • If they are like terms we can add or subtract them (depending on whether there's a \"+\" or \"-\" between them. Like logarithmic terms have the same bases and the same argument.
  • Use one of the following properties of logarithms:
    • \"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\"
    These properties require logs with the same base and coefficients of 1.
Our logs have the same bases, 2. But the arguments, x and y, are different. So we cannot add them together. Also our logs have coefficients, 5 and 1/2, which are not 1's. So we cannot use the properties... yet. Fortunately there is another property of logs, \"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\", which will help. The right side shows a log with a coefficient which is not a 1. The left side shows us how to rewrite the right side with a coefficient of 1: Just move the coefficient into the argument as its exponent!

\n" ); document.write( "So we start by using the property which lets us move the coefficients:
\n" ); document.write( "\"log%282%2C+%28x%5E5%29%29%2Blog%282%2C+%28y%5E%281%2F2%29%29%29\"
\n" ); document.write( "Since an exponent of 1/2 means square root, I'm going to rewrite it as a square root:
\n" ); document.write( "\"log%282%2C+%28x%5E5%29%29%2Blog%282%2C+%28sqrt%28y%29%29%29\"

\n" ); document.write( "Now we will use \"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\" (because its logs, like ours, have a \"+\" between them) to combine the logs:
\n" ); document.write( "\"log%282%2C+%28x%5E5%2Asqrt%28y%29%29%29\" \n" ); document.write( "
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