document.write( "Question 719905: e^x-e^-x=10 \n" ); document.write( "
Algebra.Com's Answer #441675 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"e%5Ex-e%5E%28-x%29=10\"
\n" ); document.write( "Rewriting \"e%5E%28-x%29\" as \"1%2Fe%5Ex\" will help us see what to do:
\n" ); document.write( "\"e%5Ex-1%2Fe%5Ex=10\"
\n" ); document.write( "First, we'll eliminate the fraction by multiplying both sides by \"e%5Ex\":
\n" ); document.write( "\"e%5Ex%2Ae%5Ex-e%5Ex%2A%281%2Fe%5Ex%29+=+e%5Ex%2A10\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"e%5E%282x%29-1+=+10e%5Ex\"
\n" ); document.write( "Next we'll get a zero on the right side by subtracting \"10e%5Ex\":
\n" ); document.write( "\"e%5E%282x%29-10e%5Ex-1+=+0\"
\n" ); document.write( "This does not factor, unfortunately.

\n" ); document.write( "This next part is probably the hardest part. The exponent of \"e%5E%282x%29\" is exactly twice the exponent of \"10e%5Ex\". This makes this equation an equation in what is called \"quadratic form\". Quadratic form equations can be solved in much the same way as regular quadratic equations. Using a temprary variable can help you see the \"quadratic-ness\" of our equation. Set it equal to the base and the smaller exponent:
\n" ); document.write( "Let \"q+=+e%5Ex\". Then \"q%5E2+=+%28e%5Ex%29%5E2+=+e%5E%282x%29\". Substituting these into our equation we get:
\n" ); document.write( "\"q%5E2-10q-1=0\"
\n" ); document.write( "This is clearly an quadratic equation. As noted earlier, this will not factor. But we can use the quadratic formula:
\n" ); document.write( "\"q+=+%28-%28-10%29+%2B-+sqrt%28%28-10%29%5E2-4%281%29%28-1%29%29%29%2F2%281%29\"
\n" ); document.write( "Simplifying...
\n" ); document.write( "\"q+=+%28-%28-10%29+%2B-+sqrt%28100-4%281%29%28-1%29%29%29%2F2%281%29\"
\n" ); document.write( "\"q+=+%28-%28-10%29+%2B-+sqrt%28100%2B4%29%29%2F2%281%29\"
\n" ); document.write( "\"q+=+%28-%28-10%29+%2B-+sqrt%28104%29%29%2F2%281%29\"
\n" ); document.write( "\"q+=+%2810+%2B-+sqrt%28104%29%29%2F2\"
\n" ); document.write( "\"q+=+%2810+%2B-+sqrt%284%2A26%29%29%2F2\"
\n" ); document.write( "\"q+=+%2810+%2B-+sqrt%284%29%2Asqrt%2826%29%29%2F2\"
\n" ); document.write( "\"q+=+%2810+%2B-+2%2Asqrt%2826%29%29%2F2\"
\n" ); document.write( "\"q+=+%282%285+%2B-+sqrt%2826%29%29%29%2F2\"
\n" ); document.write( "\"q+=+%28cross%282%29%285+%2B-+sqrt%2826%29%29%29%2Fcross%282%29\"
\n" ); document.write( "\"q+=+5+%2B-+sqrt%2826%29\"
\n" ); document.write( "which is short for:
\n" ); document.write( "\"q+=+5+%2B+sqrt%2826%29\" or \"q+=+5+-+sqrt%2826%29\"

\n" ); document.write( "Of course we are not interested in solutions for q. We are interested in solutions for x. So now we substitute back for the q:
\n" ); document.write( "\"e%5Ex+=+5+%2B+sqrt%2826%29\" or \"e%5Ex+=+5+-+sqrt%2826%29\"
\n" ); document.write( "We have a little more work to do to solve for x. Since \"sqrt%2826%29+%3E+5\" the second equation says that \"e%5Ex\" is a negative number. But it is impossible for e to any power to be a negative number. So that equation is impossible. There will be no solutions for x from that equation.

\n" ); document.write( "So our only solution(s) will come from the first equation. Finding the ln of each side:
\n" ); document.write( "\"ln%28e%5Ex%29+=+ln%285+%2B+sqrt%2826%29%29\"
\n" ); document.write( "On the left side we use a property of logs, \"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\", to move the exponent out in front:
\n" ); document.write( "\"x%2Aln%28e%29+=+ln%285+%2B+sqrt%2826%29%29\"
\n" ); document.write( "Since ln(e) = 1 the left side becomes:
\n" ); document.write( "\"x+=+ln%285+%2B+sqrt%2826%29%29\"
\n" ); document.write( "This is an exact expression for the solution to your equation. If you want/need a decimal approximation, get out your calculator.

\n" ); document.write( "P.S. After you have done a few of these quadratic form equations, you will not need a temporary variable. You will start to see how to go directly from:
\n" ); document.write( "\"e%5E%282x%29-10e%5Ex-1+=+0\"
\n" ); document.write( "to
\n" ); document.write( "\"e%5Ex+=+%28-%28-10%29+%2B-+sqrt%28%28-10%29%5E2-4%281%29%28-1%29%29%29%2F2%281%29\"
\n" ); document.write( "etc.
\n" ); document.write( "
\n" );