document.write( "Question 771261: Can the following equation be solved using only algebra?\r
\n" ); document.write( "\n" ); document.write( "log base 2 of (6x-8) = log base 3 of (20x+1)\r
\n" ); document.write( "\n" ); document.write( "The TI-83 gives the solution x = 4 which checks.\r
\n" ); document.write( "\n" ); document.write( "Graphing the equations y = log base 2 of (6x-8) and y = log base 3 of (20x+1) gives a solution of x = 4.\r
\n" ); document.write( "\n" ); document.write( "Can this equation be solved with algebra only? \n" ); document.write( "
Algebra.Com's Answer #470107 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The other tutor, Alan, is right about your equation.\r\n" );
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document.write( "The bases 2 and 3 are not powers of the same number. \r\n" );
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document.write( "However equations of the form \r\n" );
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document.write( "logA(Bx+C) = logD(Ex+F)\r\n" );
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document.write( "can be solved algebraically if the bases are \r\n" );
document.write( "powers of the same integer, such as this equation,\r\n" );
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document.write( "log4(2x-1) = log8(7x-8)\r\n" );
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document.write( "This is solvable by algebraic methods:\r\n" );
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document.write( "Let both sides equal to K\r\n" );
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document.write( "log4(2x-1) = K  and  log8(7x-8) = K\r\n" );
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document.write( "4K = 2x-1       and  8K = 7x-8\r\n" );
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document.write( "(22)K = 2x-1     and  (23)K = 7x-8\r\n" );
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document.write( "22K = 2x-1       and  23K = 7x-8\r\n" );
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document.write( "Raise both sides of the first equation to the 3rd power and both sides\r\n" );
document.write( "of the second equation to the 2nd power.\r\n" );
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document.write( "(22K)3 = (2x-1)3     and  (23K)2 = (7x-8)2\r\n" );
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document.write( "26K = (2x-1)3     and  26K = (7x-8)2\r\n" );
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document.write( "Now we can equate the rights sides since both equal26K\r\n" );
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document.write( "(2x-1)3 = (7x-8)2\r\n" );
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document.write( "   8x³ - 12x² + 6x - 1 = 49x²- 112x + 64\r\n" );
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document.write( "8x³ - 61x² + 118x - 65 = 0\r\n" );
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document.write( "Find possible rational zeros. \r\n" );
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document.write( "±1, \"%22%22+%2B-+1%2F2\", \"%22%22+%2B-+1%2F4\", \"%22%22+%2B-+1%2F8\", ±5, \"%22%22+%2B-+5%2F2\", \"%22%22+%2B-+5%2F4\", \"%22%22+%2B-+5%2F8\", ±13, \"%22%22+%2B-+13%2F2\", \"%22%22+%2B-+13%2F4\", \"%22%22+%2B-+13%2F8\", ±65  \r\n" );
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document.write( "Try 1 with synthetic division:\r\n" );
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document.write( "1 | 8  -61   118  -65\r\n" );
document.write( "  |      8   -53   65\r\n" );
document.write( "    8  -53    65    0\r\n" );
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document.write( "So we have factored the equation as\r\n" );
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document.write( "(x-1)(8x²-53x+65) = 0\r\n" );
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document.write( "And the quadratic factors as\r\n" );
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document.write( "(x-1)(x-5)(8x-13) = 0\r\n" );
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document.write( "x = 1, x = 5, and x = \"13%2F8\"\r\n" );
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document.write( "So there are three answers, but we must check\r\n" );
document.write( "to see if any are extraneous.  1 does not\r\n" );
document.write( "check because it causes 7x-8 to be negative.\r\n" );
document.write( "The other two solutions check.\r\n" );
document.write( "So there are two solutions: \r\n" );
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document.write( "x = 5, and x = \"13%2F8\"\r\n" );
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document.write( "But that was solvable only because the bases were both\r\n" );
document.write( "powers of 2.\r\n" );
document.write( "  | \r\n" );
document.write( "Edwin

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