document.write( "Question 1080916: Please help me solve this problem:\r
\n" ); document.write( "\n" ); document.write( "If xy = 64 and \"+log+%28x%2C+y%29+%2B+log+%28y%2C+x%29+=+5%2F2+\" find x and y. \n" ); document.write( "
Algebra.Com's Answer #695050 by ikleyn(52847)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "Please help me solve this problem:\r
\n" ); document.write( "\n" ); document.write( "If xy = 64 and \"+log+%28x%2C+y%29+%2B+log+%28y%2C+x%29+=+5%2F2+\" find x and y.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "\"log+%28x%2C+y%29+%2B+log+%28y%2C+x%29\" = \"5%2F2+\".      (1)\r\n" );
document.write( "\r\n" );
document.write( "Well known rule for the base change of logarithms says that  \"log+%28y%2C+x%29\" = \"1%2Flog%28x%2Cy%29\".\r\n" );
document.write( "\r\n" );
document.write( "Therefore, the equation (1) becomes\r\n" );
document.write( "\r\n" );
document.write( "\"log+%28x%2C+y%29\" + \"1%2Flog+%28x%2C+y%29\" = \"5%2F2+\".   (2)\r\n" );
document.write( "\r\n" );
document.write( "To solve it, introduce new variable  z = \"log+%28x%2C+y%29\".  Then the equation (2) becomes\r\n" );
document.write( "\r\n" );
document.write( "\"z+%2B+1%2Fz\" = \"5%2F2\",   or   \"z%5E2+-+%285%2F2%29z+%2B+1\" = 0,   or   \"2z%5E2+-+5z+%2B+2\" = 0.\r\n" );
document.write( "\r\n" );
document.write( "Use the quadratic formula to find its roots. The roots are  \"z%5B1%5D\" = 2   and/or  \"z%5B2%5D\" = \"1%2F2\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "1.  \"z%5B1%5D\" = 2  ====>  \"log%28x%2Cy%29\" = 2  ====>  y = x^2. \r\n" );
document.write( "\r\n" );
document.write( "    Substitute it into the equation  xy = 64,  and you will get  \"x%5E3\" = 64,   which implies  x = 4.\r\n" );
document.write( "\r\n" );
document.write( "    In this case, the solution of the original system is  x = 4,  y = 16.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2.  \"z%5B2%5D\" = \"1%2F2\"  ====>  \"log%28x%2Cy%29\" = \"1%2F2\"  ====>  y = x^(1/2). \r\n" );
document.write( "\r\n" );
document.write( "    Substitute it into the equation  xy = 64,  and you will get  x^(3/2) = 64,   which implies  \"x%5E3\" = \"64%5E2\"  and hence  x = 16.\r\n" );
document.write( "\r\n" );
document.write( "    In this case, the solution of the original system is  x = 16,  y = 4.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Answer.  There are two solutions:  1) x=4, y= 16,  and  2) x=16, y = 4.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "For properties of logarithms and solving logarithmic equations see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - WHAT IS the logarithm \r
\n" ); document.write( "\n" ); document.write( "    - Properties of the logarithm \r
\n" ); document.write( "\n" ); document.write( "    - Change of Base Formula for logarithms \r
\n" ); document.write( "\n" ); document.write( "    - Solving logarithmic equations \r
\n" ); document.write( "\n" ); document.write( "    - Using logarithms to solve real world problems \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Logarithms\".\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );