document.write( "Question 116625: Solve this equation.
\n" ); document.write( "5^x^3 = 25^x
\n" ); document.write( "x =________________(smallest value)
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\n" ); document.write( "x =__________________(largest value \n" ); document.write( "
Algebra.Com's Answer #86181 by bucky(2189)\"\" \"About 
You can put this solution on YOUR website!
In case you can still use an answer to this problem, here's how you can do it.
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\n" ); document.write( "The biggest lesson that you can learn from this problem is that anytime you see a variable used
\n" ); document.write( "as an exponent, you should think of the possibility of using logarithms to solve the problem
\n" ); document.write( "because if you take the logarithm of such a quantity, the exponent can be brought out as the
\n" ); document.write( "multiplier of the logarithm. This is true regardless of base in the logarithm. For example:
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\n" ); document.write( "\"log+7%5Ex\" is equal to \"x%2Alog+7\"
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\n" ); document.write( "In this problem you are given the equation:
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\n" ); document.write( "\"5%5E%28x%5E3%29+=+25%5Ex\"
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\n" ); document.write( "If you take the Logarithm of both sides (any base will do, but just for simplicity let's
\n" ); document.write( "use base 10), the equation becomes:
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\n" ); document.write( "\"log%2810%2C+5%5E%28x%5E3%29%29+=+log%2810%2C25%5Ex%29\"
\n" ); document.write( ".
\n" ); document.write( "bringing out the exponents and making them multipliers converts the problem to:
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\n" ); document.write( "\"x%5E3%2Alog%2810%2C5%29+=+x%2Alog%2810%2C25%29\"
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\n" ); document.write( "But notice on the right side that 25 can be replaced by \"5%5E2\" making the equation become:
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\n" ); document.write( "\"x%5E3%2Alog%2810%2C5%29+=+x%2Alog%2810%2C5%5E2%29\"
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\n" ); document.write( "Applying the exponent property again on the right side, the 2 comes out as a multiplier and
\n" ); document.write( "you have:
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\n" ); document.write( "\"x%5E3%2Alog%2810%2C5%29+=+2x%2Alog%2810%2C5%29\"
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\n" ); document.write( "Notice now that \"log%2810%2C5%29\" is a factor of both sides, and it is just a number that you
\n" ); document.write( "can get using a calculator. [Should you be interested, that number is 0.698970004].
\n" ); document.write( "If both sides of the equation are divided by \"log%2810%2C5%29\" the logarithmic term just
\n" ); document.write( "cancels on both sides and the equation is reduced to:
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\n" ); document.write( "\"x%5E3+=+2x\"
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\n" ); document.write( "Get rid of the 2x on the right side by subtracting 2x from both sides to get:
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\n" ); document.write( "\"x%5E3+-+2x+=+0\"
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\n" ); document.write( "Factor the left side and you get:
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\n" ); document.write( "\"x%2A%28x%5E2+-2%29+=+0\"
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\n" ); document.write( "This equation will be true if either of the factors on the left side equals zero because then
\n" ); document.write( "the left side would involve a multiplication by zero and that zero multiplier would make
\n" ); document.write( "the entire left side equal to zero ... therefore equaling the right side.
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\n" ); document.write( "So setting the first factor equal to zero just gives:
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\n" ); document.write( "\"x+=+0\"
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\n" ); document.write( "Then setting the second factor equal to zero gives:
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\n" ); document.write( "\"x%5E2+-+2+=+0\"
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\n" ); document.write( "Add 2 to both sides and you get:
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\n" ); document.write( "\"x%5E2+=+2\"
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\n" ); document.write( "Solve for x by taking the square root of both sides and you have:
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\n" ); document.write( "\"x+=+%2Bsqrt%282%29\" and \"x+=+-sqrt%282%29\"
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\n" ); document.write( "In summary, the three answers to this problem are:
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\n" ); document.write( "\"x+=+-sqrt%282%29\"
\n" ); document.write( "\"x+=+0\"
\n" ); document.write( "\"x+=+sqrt%282%29\"
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\n" ); document.write( "If you substitute each of these three values back into the original equation, you will find
\n" ); document.write( "that the equation is true.
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\n" ); document.write( "For example: Start with the original equation:
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\n" ); document.write( "\"5%5E%28x%5E3%29+=+25%5Ex\"
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\n" ); document.write( "Substitute 0 for x and you have:
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\n" ); document.write( "\"5%5E0+=+25%5E0\"
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\n" ); document.write( "but you know that that when any quantity is raised to the zero power the answer is 1. So this
\n" ); document.write( "equation reduces to:
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\n" ); document.write( "\"+1+=+1\"
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\n" ); document.write( "And this is obviously true ... so x = 0 is a good solution.
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\n" ); document.write( "Next start with the original equation and substitute \"sqrt%282%29\" for x.
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\n" ); document.write( "The original equation is:
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\n" ); document.write( "\"5%5E%28x%5E3%29+=+25%5Ex\"
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\n" ); document.write( "Substitute \"sqrt%282%29%29\" for x and get:
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\n" ); document.write( "\"5%5E%28%28sqrt%282%29%29%5E3%29+=+25%5E%28sqrt%282%29%29%29\"
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\n" ); document.write( "On the right side replace 25 with \"5%5E2\" and the equation becomes:
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\n" ); document.write( "\"5%5E%28%28sqrt%282%29%29%5E3%29+=+%285%5E2%29%5E%28sqrt%282%29%29\"
\n" ); document.write( ".
\n" ); document.write( "On the left side cubing \"sqrt%282%29\" is equivalent to \"%28sqrt%282%29%29%5E2%2Asqrt%282%29\" and \"%28sqrt%282%29%29%5E2+=+2\"
\n" ); document.write( "so cubing \"sqrt%282%29\" results in \"2%2Asqrt%282%29\". This makes the equation become:
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\n" ); document.write( "\"5%5E%282%2Asqrt%282%29%29+=+%285%5E2%29%5E%28sqrt%282%29%29\"
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\n" ); document.write( "Finally on the right side, the power rule of exponents tells you to multiply 2 times \"sqrt%282%29\"
\n" ); document.write( "and this makes the equation become:
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\n" ); document.write( "\"5%5E%282%2Asqrt%282%29%29+=+5%5E%282%2Asqrt%282%29%29\"
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\n" ); document.write( "Note that the left side now equals the right side, so \"x+=+sqrt%282%29\" is a good solution
\n" ); document.write( "to the problem.
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\n" ); document.write( "You can do the proof for \"x+=+-sqrt%282%29\". It works very similar to the one for \"x+=+sqrt%282%29\".
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\n" ); document.write( "I know this is late, but the lessons involved in solving this problem are hopefully in
\n" ); document.write( "time to help you understand the concepts involved in getting a solution.
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\n" ); document.write( "Happy holidays to you and yours ... \n" ); document.write( "
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