SOLUTION: Solve: 16^(x) + 4^(x+1) - 3 = 0 Thanks!

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Question 913163: Solve: 16^(x) + 4^(x+1) - 3 = 0
Thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
16%5E%28x%29+%2B+4%5E%28x%2B1%29+-+3+=+0

%284%5E2%29%5Ex+%2B+4%5E%28x%2B1%29+-+3+=+0

4%5E%282x%29+%2B+4%5E%28x%2B1%29+-+3+=+0

%284%5Ex%29%5E2+%2B+4%5E%28x%2B1%29+-+3+=+0

%284%5Ex%29%5E2+%2B+%284%5Ex%29%2A%284%5E%281%29%29+-+3+=+0

%284%5Ex%29%5E2+%2B+%284%5Ex%29%2A%284%29+-+3+=+0

%284%5Ex%29%5E2+%2B+4%284%5Ex%29+-+3+=+0

z%5E2+%2B4z+-+3+=+0 Let z+=+4%5Ex

Use the quadratic formula to get z+=+-2%2Bsqrt%287%29 or z+=+-2-sqrt%287%29

The steps for the quadratic formula are lengthy and I feel they would clutter up this solution page. Let me know if you need to see the steps to getting those solutions for z.

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The solutions in terms of z are z+=+-2%2Bsqrt%287%29 or z+=+-2-sqrt%287%29

We ultimately want solutions in terms of x.

Plug in the first solution for z to get the first solution for x.

z+=+4%5Ex

-2%2Bsqrt%287%29+=+4%5Ex

4%5Ex+=+-2%2Bsqrt%287%29

x+=+log%284%2C%28-2%2Bsqrt%287%29%29%29

So the first solution is x+=+log%284%2C%28-2%2Bsqrt%287%29%29%29

Using a calculator, we get log(-2+sqrt(7))/log(4) = -0.3154747

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Plug in the second solution for z to get the second solution for x.

z+=+4%5Ex

-2-sqrt%287%29+=+4%5Ex

4%5Ex+=+-2-sqrt%287%29

x+=+log%284%2C%28-2-sqrt%287%29%29%29

So the second solution is x+=+log%284%2C%28-2-sqrt%287%29%29%29; however, notice how -2-sqrt%287%29=-4.64575 is negative. That means the log of this value is undefined. So log(-2-sqrt(7)) = log(-4.64575) = Undefined

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Final Answer:

The only exact solution is x+=+log%284%2C%28-2%2Bsqrt%287%29%29%29

The only approximate solution is x+=+-0.3154747


Let me know if you need more help or if you need me to explain a step in more detail.
Feel free to email me at jim_thompson5910@hotmail.com
or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html

Thanks,

Jim