document.write( "Question 354508: Write log a sqrt(6) , in terms of \"p\" and \"q\" , if log a 2 = p and log a 3 = q\r
\n" ); document.write( "\n" ); document.write( "Thank you for taking the time to help me out.
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Algebra.Com's Answer #253325 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"log%28a%2C+%282%29%29+=+p\"
\n" ); document.write( "\"log%28a%2C+%283%29%29+=+q\"
\n" ); document.write( "\"log%28a%2C+%28sqrt%286%29%29%29\"
\n" ); document.write( "The first key to this problem is to know that square roots can be expressed using fractional exponents: \"sqrt%28x%29+=+x%5E%281%2F2%29\". So we can rewrite your expression using an exponent of 1/2:
\n" ); document.write( "\"log%28a%2C+%286%5E%281%2F2%29%29%29%29\"
\n" ); document.write( "Now we can start using the properties of logarithms:
  • \"log%28n%2C+%28x%2Ay%29%29+=+log%28n%2C+%28x%29%29+%2B+log%28n%2C+%28y%29%29\"
  • \"log%28n%2C+%28x%2Fy%29%29+=+log%28n%2C+%28x%29%29+-+log%28n%2C+%28y%29%29\"
  • \"log%28n%2C+%28x%5Ey%29%29+=+y%2Alog%28n%2C+%28x%29%29\"

\n" ); document.write( "We can use the third property to remove the exponent:
\n" ); document.write( "\"%281%2F2%29%2Alog%28a%2C+%286%29%29\"
\n" ); document.write( "Now we need to change the argument from a six to something in terms of 2's and/or 3's. It should be obvious that 6 = 2*3 so now we have:
\n" ); document.write( "\"%281%2F2%29%2Alog%28a%2C+%282%2A3%29%29\"
\n" ); document.write( "In this form we can use the first property to separate the log into two logs:
\n" ); document.write( "\"%281%2F2%29%2A%28log%28a%2C+%282%29%29+%2B+log%28a%2C+%283%29%29%29\"
\n" ); document.write( "Next we can subsitute in for the two logs:
\n" ); document.write( "\"%281%2F2%29%2A%28p+%2B+q%29\"
\n" ); document.write( "And finally we can use the Distributive Property:
\n" ); document.write( "\"%281%2F2%29%2Ap+%2B+%281%2F2%29%2Aq\" \n" ); document.write( "
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