document.write( "Question 1209850: Given that log_{4n} 40 sqrt(3) = log_{2n} 10, find n^4. \n" ); document.write( "
Algebra.Com's Answer #850883 by ikleyn(52824)\"\" \"About 
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\n" ); document.write( "Given that log_{4n} 40 sqrt(3) = log_{2n} 10, find n^4.
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document.write( "You are given  \r\n" );
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document.write( "    \"log%28%284n%29%2C+%2840%2Asqrt%283%29%29%29\" = \"log%28%282n%29%2C%2810%29%29\".    (1)\r\n" );
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document.write( "Let x be the common numerical value of each side of (1), so we can write\r\n" );
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document.write( "    \"log%28%284n%29%2C+%2840%2Asqrt%283%29%29%29\" = x,    (2)\r\n" );
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document.write( "    \"log%28%282n%29%2C%2810%29%29\"    = x.    (3)\r\n" );
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document.write( "Then we can rewrite (2) and (3) in this form, respectively\r\n" );
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document.write( "    \"%284n%29%5Ex\" = \"40%2Asqrt%283%29\",        (2')\r\n" );
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document.write( "    \"%282n%29%5Ex\" = 10.            (3')\r\n" );
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document.write( "Divide equation (2') by equation (3').  You will get\r\n" );
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document.write( "    \"%284n%29%5Ex%2F%282n%29%5Ex\" = \"4%2Asqrt%283%29\".\r\n" );
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document.write( "You can simplify left side\r\n" );
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document.write( "    \"2%5Ex\" = \"4%2Asqrt%283%29\".\r\n" );
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document.write( "Hence,\r\n" );
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document.write( "    x = \"log%282%2C%284%2Asqrt%283%29%29%29\".\r\n" );
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document.write( "        At this point, we solved half of the problem \r\n" );
document.write( "        and have found the numerical value of each \r\n" );
document.write( "        expression in the left side of equations (2) and (3).\r\n" );
document.write( "        Now we are going to make next step and to find n and \"n%5E4\".\r\n" );
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document.write( "We can write equation (3) in the form\r\n" );
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document.write( "    \"log%28%282n%29%2C%2810%29%29\"    =  \"log%282%2C%284%2Asqrt%283%29%29%29\".    (4)\r\n" );
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document.write( "Let y be the common numerical value of each side of (4), so we can write\r\n" );
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document.write( "    \"log%28%282n%29%2C%2810%29%29\" = y,                  (5)\r\n" );
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document.write( "    \"log%282%2C%284%2Asqrt%283%29%29%29\" = y.                   (6)\r\n" );
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document.write( "Then we can rewrite (5) and (6) in this form, respectively\r\n" );
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document.write( "    \"%282n%29%5Ey\" = 10,                       (5')\r\n" );
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document.write( "    \"2%5Ey\" = \"4%2Asqrt%283%29\".                      (6')\r\n" );
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document.write( "Divide equation (5') by equation (6').  You will get\r\n" );
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document.write( "    \"n%5Ey\" = \"10%2F%284%2Asqrt%283%29%29\" = \"5%2F%282%2Asqrt%283%29%29\" = \"%285%2Asqrt%283%29%29%2F6\".\r\n" );
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document.write( "Take logarithm of both sides\r\n" );
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document.write( "    y*log(n) = \"log%28%28%285%2Asqrt%283%29%29%2F6%29%29%29\".    (7)\r\n" );
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document.write( "In this formula, it does not matter, which base of logarithm I use.  \r\n" );
document.write( "Let's the base of logarithm in this equation be 10.\r\n" );
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document.write( "From equation (7), \r\n" );
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document.write( "    log(n) = \"%281%2Fy%29%2Alog%28%28%285%2Asqrt%283%29%29%2F6%29%29\".\r\n" );
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document.write( "We can substitute here expression (6) for y.  We get then\r\n" );
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document.write( "    log(n) = \"1%2Flog%282%2C%284%2Asqrt%283%29%29%29\" * \"log%28%28%285%2Asqrt%283%29%29%2F6%29%29\".\r\n" );
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document.write( "This is the  \"precise\"  expression for log(n).\r\n" );
document.write( "We can find its numerical value \r\n" );
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document.write( "    log(n) = \"0.159379381%2F2.79248125\" = 0.057074.\r\n" );
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document.write( "Hence,  n = \"10%5E0.057074\" = 1.140444,  and  \"n%5E4\" = \"1.140444%5E4\" = 1.691601, approximately.\r\n" );
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document.write( "At this point, the solution is complete and the value of  \"n%5E4\"  is found.\r\n" );
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document.write( "ANSWER.  \"n%5E4\" = \"1.140444%5E4\" = 1.691601,  approximately.\r\n" );
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\n" ); document.write( "\n" ); document.write( "After completing this solution, I checked equation (1)
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