document.write( "Question 1144754: Find x,
\n" ); document.write( "4^(x+1.5) + 9^(x+0.5) = 10×6^x \n" ); document.write( "
Algebra.Com's Answer #765951 by ikleyn(52788)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            It is an advanced level exponential equation.\r
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\n" ); document.write( "\n" ); document.write( "            Nevertheless,  it is  SOLVABLE  (!),  and I will show you how.\r
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document.write( "4^(x+1.5) + 9^(x+0.5) = 10×6^x             (1)\r\n" );
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document.write( "It is equivalent to\r\n" );
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document.write( "2^(2x+3) + 3^(2x+1) = 10*(2^x)*(3^x),   or\r\n" );
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document.write( "8*2^(2x) + 3*3^(2x) = 10*(2^x)*(3^x).      (2) \r\n" );
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document.write( "Introduce new variables  u = 2^x,  v = 3^x.  Then equation (2) takes the form\r\n" );
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document.write( "    8u^2 + 3v^2 = 10uv\r\n" );
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document.write( "Divide both sides by v^2.  You will get\r\n" );
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document.write( "    \"8%2A%28u%2Fv%29%5E2\" + 3 - \"10%2A%28u%2Fv%29\" = 0                (3)\r\n" );
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document.write( "Let z = \"u%2Fv\".   Then equation (3) takes the form\r\n" );
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document.write( "    8z^2 - 10z + 3 = 0.\r\n" );
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document.write( "Solve this quadratic equation using the quadratic formula\r\n" );
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document.write( "    \"z%5B1%2C2%5D\" = \"%2810+%2B-+sqrt%2810%5E2+-+4%2A8%2A3%29%29%2F%282%2A8%29\" = \"%2810+%2B-+2%29%2F16\".\r\n" );
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document.write( "The roots are  \r\n" );
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document.write( "    \"z%5B1%5D\" = \"%2810%2B2%29%2F16\" = \"12%2F16\" = \"3%2F4\",   and\r\n" );
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document.write( "    \"z%5B2%5D\" = \"%2810-2%29%2F16\" = \"8%2F16\" = \"1%2F2\".\r\n" );
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document.write( "Thus, we should consider two cases.\r\n" );
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document.write( "(a)  \"z%5B1%5D\" = \"3%2F4\".\r\n" );
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document.write( "     It means  \"3%2F4\" = \"u%2Fv\" = \"%282%2F3%29%5Ex\".\r\n" );
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document.write( "     Next you can take any logarithm, log base 10, or natural logarithm \"ln\" to continue\r\n" );
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document.write( "       \"log%2810%2C%283%2F4%29%29\" = \"x%2A%28log%2810%2C%282%2F3%29%29%29\",\r\n" );
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document.write( "       \"x%2A%28log%282%29+-+log%283%29%29\" = \"log%283%29+-+log%284%29\"\r\n" );
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document.write( "        x = \"%28log%283%29+-+2%2Alog%282%29%29%2F%28log%282%29+-+log%283%29%29\" = \"%282%2Alog%282%29+-+log%283%29%29%2F%28log%283%29+-+log%282%29%29\" = 0.709511  (approximately).\r\n" );
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document.write( "The first case is completed.\r\n" );
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document.write( "Similarly, you can consider and complete the second case  \"z%5B2%5D\" = \"1%2F2\".\r\n" );
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document.write( "May I leave it to you, in order for you completed it on your own ?\r\n" );
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\n" ); document.write( "\n" ); document.write( "If you still will have questions, let me know.\r
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\n" ); document.write( "\n" ); document.write( "            Come again to this forum soon to learn something new  (!)\r
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document.write( "As an accurate person, I checked my answer.\r\n" );
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document.write( "I used MS Excel in my computer and calculated the left side and the right side expressions of the original equation at x= 0.709511.\r\n" );
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document.write( "I got the value of  35.65389  on the left side and the value of  35.65389  on the right side.\r\n" );
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