document.write( "Question 170138This question is from textbook intermediate algebra
\n" ); document.write( ": 90) 3^2x -8 . 3^x +15=0
\n" ); document.write( "94) If x= (log125 5)^log5 125\r
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Algebra.Com's Answer #125568 by jim_thompson5910(35256)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "Problem #90
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\n" ); document.write( "\n" ); document.write( "\"3%5E%282x%29-8%2A3%5E%28x%29%2B15=0\" Start with the given equation\r
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\n" ); document.write( "\n" ); document.write( "Let \"z=3%5E%28x%29\". So this means that \"z%5E2=%283%5E%28x%29%29%5E2=3%5E%282x%29\" (ie \"z=3%5E%282x%29\")\r
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\n" ); document.write( "\n" ); document.write( "\"z%5E2-8z%2B15=0\" Replace \"3%5E%282x%29\" with \"z%5E2\" and \"3%5E%28x%29\" with \"z\" (see above)\r
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\n" ); document.write( "\n" ); document.write( "Notice we have a quadratic equation in the form of \"az%5E2%2Bbz%2Bc\" where \"a=1\", \"b=-8\", and \"c=15\"\r
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\n" ); document.write( "\n" ); document.write( "Let's use the quadratic formula to solve for z\r
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\n" ); document.write( "\n" ); document.write( "\"z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\" Start with the quadratic formula\r
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\n" ); document.write( "\n" ); document.write( "\"z+=+%28-%28-8%29+%2B-+sqrt%28+%28-8%29%5E2-4%281%29%2815%29+%29%29%2F%282%281%29%29\" Plug in \"a=1\", \"b=-8\", and \"c=15\"\r
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\n" ); document.write( "\n" ); document.write( "\"z+=+%288+%2B-+sqrt%28+%28-8%29%5E2-4%281%29%2815%29+%29%29%2F%282%281%29%29\" Negate \"-8\" to get \"8\". \r
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\n" ); document.write( "\n" ); document.write( "\"z+=+%288+%2B-+sqrt%28+64-4%281%29%2815%29+%29%29%2F%282%281%29%29\" Square \"-8\" to get \"64\". \r
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\n" ); document.write( "\n" ); document.write( "\"z+=+%288+%2B-+sqrt%28+64-60+%29%29%2F%282%281%29%29\" Multiply \"4%281%29%2815%29\" to get \"60\"\r
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\n" ); document.write( "\n" ); document.write( "\"z+=+%288+%2B-+sqrt%28+4+%29%29%2F%282%281%29%29\" Subtract \"60\" from \"64\" to get \"4\"\r
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\n" ); document.write( "\n" ); document.write( "\"z+=+%288+%2B-+sqrt%28+4+%29%29%2F%282%29\" Multiply \"2\" and \"1\" to get \"2\". \r
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\n" ); document.write( "\n" ); document.write( "\"z+=+%288+%2B-+2%29%2F%282%29\" Take the square root of \"4\" to get \"2\". \r
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\n" ); document.write( "\n" ); document.write( "\"z+=+%288+%2B+2%29%2F%282%29\" or \"z+=+%288+-+2%29%2F%282%29\" Break up the expression. \r
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\n" ); document.write( "\n" ); document.write( "\"z+=+%2810%29%2F%282%29\" or \"z+=++%286%29%2F%282%29\" Combine like terms. \r
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\n" ); document.write( "\n" ); document.write( "\"z+=+5\" or \"z+=+3\" Simplify. \r
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\n" ); document.write( "\n" ); document.write( "Now remember, we let \"z=3%5E%28x%29\". So this means that \"3%5E%28x%29+=+5\" or \"3%5E%28x%29+=+3\"\r
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\n" ); document.write( "\n" ); document.write( "\"3%5E%28x%29+=+5\" Start with the first equation\r
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\n" ); document.write( "\n" ); document.write( "\"x+=+log%283%2C%285%29%29\" Take the log base 3 of both sides (to isolate \"x\")\r
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\n" ); document.write( "\n" ); document.write( "\"3%5E%28x%29+=+3\" Move onto the second equation\r
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\n" ); document.write( "\n" ); document.write( "\"x+=+log%283%2C%283%29%29\" Take the log base 3 of both sides (to isolate \"x\")\r
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\n" ); document.write( "\n" ); document.write( "\"x+=+1\" Evaluate the log base 3 of 3 to get 1\r
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\n" ); document.write( "\n" ); document.write( "So the solutions are \"x+=+log%283%2C%285%29%29\" or \"x+=1\" \r
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\n" ); document.write( "\n" ); document.write( "Is the equation \"x=%28log%28125%2C%285%29%29%29%5E%28log%285%2C%28125%29%29%29\"????\r
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\n" ); document.write( "\n" ); document.write( "If so, then....\r
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\n" ); document.write( "\n" ); document.write( "\"x=%28log%28125%2C%285%29%29%29%5E%28log%285%2C%28125%29%29%29\" Start with the given equation\r
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\n" ); document.write( "\n" ); document.write( "Let \"w=log%28125%2C%285%29%29\" and \"z=log%285%2C%28125%29%29\"\r
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\n" ); document.write( "\n" ); document.write( "So the equation then becomes...\r
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\n" ); document.write( "\n" ); document.write( "\"x=%28w%29%5E%28z%29\" \r
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\n" ); document.write( "\n" ); document.write( "Now let's evaluate the individual pieces:\r
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\n" ); document.write( "\n" ); document.write( "\"w=log%28125%2C%285%29%29\" Start with the first assignment\r
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\n" ); document.write( "\n" ); document.write( "\"w=log%2810%2C%285%29%29%2Flog%2810%2C%28125%29%29\" Use the change of base formula to break up the log\r
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\n" ); document.write( "\n" ); document.write( "\"w=log%2810%2C%285%29%29%2Flog%2810%2C%285%5E3%29%29\" Rewrite 125 as \"5%5E3\". Note: \"5%5E3=125\"\r
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\n" ); document.write( "\n" ); document.write( "\"w=log%2810%2C%285%29%29%2F%283%2Alog%2810%2C%285%29%29%29\" Pull the exponent down and place it out front.\r
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\n" ); document.write( "\n" ); document.write( "\"w=1%2F3\" Divide and cancel out the common terms.\r
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\n" ); document.write( "\n" ); document.write( "Since \"w=1%2F3\", this means that \"log%28125%2C%285%29%29=1%2F3\" (this is the first piece)\r
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\n" ); document.write( "\n" ); document.write( "\"z=log%285%2C%28125%29%29\" Now move onto the second assignment\r
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\n" ); document.write( "\n" ); document.write( "\"z=log%2810%2C%28125%29%29%2Flog%2810%2C%285%29%29\" Use the change of base formula to break up the log\r
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\n" ); document.write( "\n" ); document.write( "\"z=log%2810%2C%285%5E3%29%29%2Flog%2810%2C%285%29%29\" Rewrite 125 as \"5%5E3\". Note: \"5%5E3=125\"\r
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\n" ); document.write( "\n" ); document.write( "\"z=%283%2Alog%2810%2C%285%29%29%29%2Flog%2810%2C%285%29%29\" Pull down the exponent.\r
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\n" ); document.write( "\n" ); document.write( "\"z=3\" Divide and cancel out the common terms.\r
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\n" ); document.write( "\n" ); document.write( "Since \"z=3\", this means that \"log%285%2C%28125%29%29=3\" (which is the second piece)\r
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\n" ); document.write( "\n" ); document.write( "\"x=w%5Ez\" Go back to the original equation\r
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\n" ); document.write( "\n" ); document.write( "\"x=%281%2F3%29%5E3\" Plug in \"w=1%2F3\" and \"z=3\"\r
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\n" ); document.write( "\n" ); document.write( "\"x=%281%5E3%29%2F%283%5E3%29\" Distribute the exponent.\r
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\n" ); document.write( "\n" ); document.write( "\"x=1%2F27\" Cube 1 to get 1. Cube 3 to get 27. \r
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\n" ); document.write( "\n" ); document.write( "So the answer is \"x=1%2F27\"\r
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\n" ); document.write( "\n" ); document.write( "This means that \"%28log%28125%2C%285%29%29%29%5E%28log%285%2C%28125%29%29%29=1%2F27\" \n" ); document.write( "
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