SOLUTION: 2^x - 6(2^-x) = 6 This problem is from Schaum's Outlines - Precalculus, chapter 19 page 180. That book doesn't provide many examples (let alone good ones) so, while I can solv

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: 2^x - 6(2^-x) = 6 This problem is from Schaum's Outlines - Precalculus, chapter 19 page 180. That book doesn't provide many examples (let alone good ones) so, while I can solv      Log On


   

Question 115120: 2^x - 6(2^-x) = 6
This problem is from Schaum's Outlines - Precalculus, chapter 19 page 180.
That book doesn't provide many examples (let alone good ones) so, while I can solve a similar one (or that same one if I plug = 1 instead of = 6), I don't know how to solve this one.
It should be ??? = 3 and then we work it out, but unless I'm muliplying 2^x by 1, I don't know what u will equal. So I can't find the value for x.
(2^x)^2 - 6 = 6.2x
u^2 = (2^x)^2 u = ???

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
2%5Ex+-+6%2A2%5E%28-x%29+=+6 Start with the given equation


2%5Ex+-+6%2A%28%282%5Ex%29%5E%28-1%29%29+=+6 Write 2%5E%28-x%29 as 2%5E%28-1%2Ax%29=%282%5Ex%29%5E%28-1%29


Now let u=2%5Ex

u+-+6u%5E%28-1%29+=+6


u+-+6%281%2Fu%29+=+6 Rewrite u%5E%28-1%29 as 1%2Fu


u%28u+-+6%281%2Fu%29%29+=u%2A+6 Multiply both sides by the LCD u


u%5E2+-+6+=6u Distribute and multiply

u%5E2-6u-6=0 Subtract 6u from both sides



Now let's use the quadratic formula to solve for u:


Starting with the general quadratic

au%5E2%2Bbu%2Bc=0

the general solution using the quadratic equation is:

u+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve u%5E2-6%2Au-6=0 ( notice a=1, b=-6, and c=-6)




u+=+%28--6+%2B-+sqrt%28+%28-6%29%5E2-4%2A1%2A-6+%29%29%2F%282%2A1%29 Plug in a=1, b=-6, and c=-6



u+=+%286+%2B-+sqrt%28+%28-6%29%5E2-4%2A1%2A-6+%29%29%2F%282%2A1%29 Negate -6 to get 6



u+=+%286+%2B-+sqrt%28+36-4%2A1%2A-6+%29%29%2F%282%2A1%29 Square -6 to get 36 (note: remember when you square -6, you must square the negative as well. This is because %28-6%29%5E2=-6%2A-6=36.)



u+=+%286+%2B-+sqrt%28+36%2B24+%29%29%2F%282%2A1%29 Multiply -4%2A-6%2A1 to get 24



u+=+%286+%2B-+sqrt%28+60+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



u+=+%286+%2B-+2%2Asqrt%2815%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



u+=+%286+%2B-+2%2Asqrt%2815%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

u+=+%286+%2B+2%2Asqrt%2815%29%29%2F2 or u+=+%286+-+2%2Asqrt%2815%29%29%2F2


Now break up the fraction


u=%2B6%2F2%2B2%2Asqrt%2815%29%2F2 or u=%2B6%2F2-2%2Asqrt%2815%29%2F2


Simplify


u=3%2Bsqrt%2815%29 or u=3-sqrt%2815%29


So these expressions approximate to

u=6.87298334620742 or u=-0.872983346207417


So our possible solutions are:
u=6.87298334620742 or u=-0.872983346207417



Now remember, we let u=2%5Ex. So this means (I solved for x)

Since we cannot take the log of a negative number, we must discard the second solution. Now lets find x when u=6.87298334620742


Start with the given equation


x=log%282%2C%286.87298334620742%29%29 Plug in u=6.87298334620742


x=2.78093646360161 Evaluate the expression using a calculator


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Answer:
So our answer is approximately x=2.78