document.write( "Question 54932: Please help me wit this problem:
\n" ); document.write( "Find the exact solution to the equation 3^x+5=9^x \n" ); document.write( "

Algebra.Com's Answer #37147 by josmiceli(19441) $\"\"$ $\"About$

You can put this solution on YOUR website!
This might not be the shortest solution, but it seems to check OK
\n" ); document.write( " $\"3%5Ex%2B5=9%5Ex\"$
\n" ); document.write( "note that $\"9%5Ex+=+%283%5E2%29%5Ex\"$ , so
\n" ); document.write( " $\"3%5Ex+%2B+5+=+%283%5E2%29%5Ex\"$
\n" ); document.write( "also, $\"%283%5E2%29%5Ex+=+3%5E%282x%29\"$ because, if x were 5, let's say, then
\n" ); document.write( "

\n" ); document.write( " $\"3%5Ex+%2B+5+=+3%5E%282x%29\"$
\n" ); document.write( "subtract 3^x from both sides
\n" ); document.write( " $\"3%5E%282x%29+-+3%5Ex+=+5\"$
\n" ); document.write( "note that $\"3%5E%282x%29+=+3%5E%28x%2Bx%29+=+%283%5Ex%29%2A%283%5Ex%29\"$
\n" ); document.write( " $\"%283%5Ex%29%2A%283%5Ex%29+-+3%5Ex+=+5\"$
\n" ); document.write( "now factor out 3^x
\n" ); document.write( " $\"%283%5Ex%29%2A%283%5Ex+-+1%29+=+5\"$
\n" ); document.write( "make a substitution, let $\"z+=+3%5Ex\"$ , so
\n" ); document.write( " $\"z%28z+-+1%29+=+5\"$
\n" ); document.write( " $\"z%5E2+-+z+=+5\"$
\n" ); document.write( "solve completing the square
\n" ); document.write( " $\"z%5E2+-+z+%2B+1%2F4+=+5+%2B+1%2F4\"$
\n" ); document.write( " $\"z+-+1%2F2%29%5E2+=+21%2F4\"$
\n" ); document.write( "take the square root of both sides
\n" ); document.write( " $\"z+-+1%2F2+=+sqrt%280+%2B+21%29%2F2\"$
\n" ); document.write( " $\"z+=+%281+%2B+sqrt%280+%2B+21%29%29+%2F+2\"$
\n" ); document.write( " $\"z+=+5.582576+%2F+2\"$
\n" ); document.write( " $\"z+=+2.7912878\"$
\n" ); document.write( "since $\"z+=+3%5Ex\"$
\n" ); document.write( " $\"3%5Ex+=+2.7912878\"$
\n" ); document.write( "take the log of both sides. I chose log to the base e.
\n" ); document.write( " $\"x%2Aln%283%29+=+1.0265031\"$
\n" ); document.write( " $\"1.0986123%2Ax+=+1.0265031\"$
\n" ); document.write( " $\"x+=+1.0265031+%2F+1.0986123\"$
\n" ); document.write( " $\"x+=+.9343634\"$ answer
\n" ); document.write( "I'll sub this back into the original equation
\n" ); document.write( " $\"3%5Ex%2B5=9%5Ex\"$
\n" ); document.write( " $\"3%5E.9343634+=+2.7912879\"$
\n" ); document.write( " $\"9%5E.9343634+=+7.7912883\"$
\n" ); document.write( " $\"+2.7912879+%2B+5+=+7.7912883\"$
\n" ); document.write( "This is true up to the 5th decimal place, so I believe the error is due
\n" ); document.write( "to my rounding off on calculator
\n" ); document.write( " Note also that I could have used the negative square root of 21
\n" ); document.write( "in the calculation for z, but this would have made $\"3%5Ex\"$ negative,
\n" ); document.write( "and that's impossible, no matter what x is. \n" ); document.write( "

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