document.write( "Question 727931: log3x + logx9 = 3. Find x. My answer is 3. \n" ); document.write( "
Algebra.Com's Answer #445247 by Edwin McCravy(20081)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "There are TWO solutions for x, not just one.  To do this you\r\n" );
document.write( "need to know the change of base formula \"log%28b%2Ca%29+=+log%28c%2Ca%29%2Flog%28c%2Cb%29\"\r\n" );
document.write( "\r\n" );
document.write( "and the definition of a logarithm:  \"log%28b%2Ca%29=c\" where c is the exponent to\r\n" );
document.write( "which base b must be raised to give a.  That is \"log%28b%2Ca%29=c\" is equivalent\r\n" );
document.write( "to the equation \"a=b%5Ec\"  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "log3x + logx9 = 3\r\n" );
document.write( "\r\n" );
document.write( "Use the change of base formula on the second term:\r\n" );
document.write( "\r\n" );
document.write( "log3x + \"log%283%2C9%29%2Flog%283%2Cx%29\" = 3\r\n" );
document.write( "\r\n" );
document.write( "Multiply through by log3x\r\n" );
document.write( "\r\n" );
document.write( "(log3x)² + log39 = 3·log3x\r\n" );
document.write( "\r\n" );
document.write( "log39 = 2 since the exponent to which the base 3 must be\r\n" );
document.write( "raised to give 9 is the exponent 2, since 3² = 9 \r\n" );
document.write( "\r\n" );
document.write( "(log3x)² + 2 = 3·log3x\r\n" );
document.write( "\r\n" );
document.write( "(log3x)² - 3·log3x + 2 = 0\r\n" );
document.write( "\r\n" );
document.write( "That factors as a quadratic in log3x\r\n" );
document.write( "\r\n" );
document.write( "(log3x - 2)(log3x - 1) = 0\r\n" );
document.write( "\r\n" );
document.write( "Use the zero-factor property:\r\n" );
document.write( "\r\n" );
document.write( "log3x - 2 = 0;   log3x - 1 = 0;\r\n" );
document.write( "    log3x = 2;       log3x = 1;\r\n" );
document.write( "        x = 32          x = 31\r\n" );
document.write( "        x = 9           x = 3\r\n" );
document.write( "\r\n" );
document.write( "Two solutions: 9 and 3.\r\n" );
document.write( "\r\n" );
document.write( "Checking x = 9\r\n" );
document.write( "\r\n" );
document.write( "log3x + logx9 = 3\r\n" );
document.write( "log39 + log99 = 3\r\n" );
document.write( "  2 + 1 = 3\r\n" );
document.write( "      3 = 3\r\n" );
document.write( "\r\n" );
document.write( "Checking x = 3\r\n" );
document.write( "\r\n" );
document.write( "log3x + logx9 = 3\r\n" );
document.write( "log33 + log39 = 3\r\n" );
document.write( "  1 + 2 = 3\r\n" );
document.write( "      3 = 3\r\n" );
document.write( "\r\n" );
document.write( "Both answers check.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "
\n" );