Algebra.Com's Answer #841044 by ikleyn(52909)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "Suppose  , where  .
\n" );
document.write( "Compute the pair of positive integers (b,n) that satisfies this equation,
\n" );
document.write( "where b is the minimum value greater than 10.
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "The starting equation is  =  . (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Take log base 11 of both sides. \r\n" );
document.write( "Using basic elementary properties of logarithms, you will get\r\n" );
document.write( "\r\n" );
document.write( "  =  . (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Cancel common factor  in both sides. You will get\r\n" );
document.write( "\r\n" );
document.write( "  =  . (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Using  =  , rewrite equation (3) in equivalent form\r\n" );
document.write( "\r\n" );
document.write( "  =  . (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At this point, you have logarithms with common (the same) base 56.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Rewrite proportion (4) in equivalent form\r\n" );
document.write( "\r\n" );
document.write( "  =  . (5)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "According to the \"base change\" formula for logarithms, the right side of (5) is\r\n" );
document.write( "\r\n" );
document.write( " =  . (6)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, proportion (5) is now this significantly simplified equation\r\n" );
document.write( "\r\n" );
document.write( " = . (7)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Rewrite it equivalently in this form\r\n" );
document.write( "\r\n" );
document.write( " x =  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Again, apply the formula  =  to get\r\n" );
document.write( "\r\n" );
document.write( " x =  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "But they want x =  (see the description of the problem).\r\n" );
document.write( "\r\n" );
document.write( "It leads us to this equation\r\n" );
document.write( "\r\n" );
document.write( " = ,\r\n" );
document.write( "\r\n" );
document.write( "which is the same as\r\n" );
document.write( "\r\n" );
document.write( " =  . (8)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From (8), we get the answer to the problem: b = 13; n =  = 1331.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER. The solution to the problem is b = 13; n = = 1331.\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Solved.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "///////////////////////\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "In students' jargon, this problem is a coffin, that is, solving it seems impossible.\r
\n" );
document.write( "\n" );
document.write( "However, at this forum you can get a solution even for such a grave task.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Do not forget to post your \" THANKS \" to me for my teaching.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "When you will post your \" THANKS \", please refer to the ID number
\n" );
document.write( "of this problem, which is 1204647.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |