document.write( "Question 773512: What is the answer to 3(7^2x) + 7^x+1 - 6 = 0? \n" ); document.write( "

Algebra.Com's Answer #471720 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
I read what you wrote as $\"3%287%5E2x%29+%2B+7%5Ex%2B1+-+6+=+0\"$ or maybe $\"3%2A7%5E%282x%29+%2B+7%5Ex%2B1+-+6+=+0\"$
\n" ); document.write( "I am going to assume that you meant $\"3%2A7%5E%282x%29+%2B+7%5E%28x%2B1%29+-+6+=+0\"$
\n" ); document.write( "I would change variables, defining $\"y=7%5Ex\"$ .
\n" ); document.write( " $\"y=7%5Ex\"$ means $\"y%5E2=%287%5Ex%29%5E2=7%5E%282x%29\"$
\n" ); document.write( "Also, $\"7%5E%28x%2B1%29=7%5Ex%2A7%5E1=7%2A7%5Ex\"$
\n" ); document.write( "Substituting all those changes the equation becomes
\n" ); document.write( " $\"3y%5E2%2B7y-6=0\"$
\n" ); document.write( "Factoring we transform that equation into
\n" ); document.write( " $\"%283y-2%29%28y%2B3%29=0\"$
\n" ); document.write( "with solutions $\"y=2%2F3\"$ and $\"y=-3\"$ .
\n" ); document.write( "You could get the same solutions using the quadratic formula, if you do not like factoring.
\n" ); document.write( "Going bacl to $\"x\"$ as a variable, we see that
\n" ); document.write( " $\"y=-3\"$ would mean $\"7%5Ex=-3\"$ and that is impossible,
\n" ); document.write( "so we are left with
\n" ); document.write( " $\"y=2%2F3\"$ which means $\"7%5Ex=2%2F3\"$
\n" ); document.write( "Now we take logarithms of both sides.
\n" ); document.write( "If all we wanted is an exact solution, we could use base 7 logarithms:
\n" ); document.write( " $\"log%287%2C7%5Ex%29=log%287%2C2%2F3%29\"$ --> $\"x=log%287%2C2%2F3%29\"$ <--> $\"x=log%287%2C2%29-log%287%2C3%29\"$
\n" ); document.write( "However, if we wanted to get an approximate solution, we would use the more common base 10 logs.
\n" ); document.write( "Probably that is what the teacher expects. It would be
\n" ); document.write( " $\"log%28%287%5Ex%29%29=log%28%282%2F3%29%29\"$ --> $\"x%2Alog%28%287%29%29=log%28%282%2F3%29%29\"$ --> $\"highlight%28x=log%28%28%222%2F3%22%29%29%2Flog%28%287%29%29%29\"$ or $\"highlight%28x=%28log%28%282%29%29-log%28%283%29%29%29%2Flog%28%287%29%29%29\"$ \n" ); document.write( "

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