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omeli [17]
3 years ago
15

How do I solve for x: (4/9)^(x) • (8/27)^(1-x) =2/3

Mathematics
2 answers:
asambeis [7]3 years ago
8 0

(\frac{4}{9})^x\cdot (\frac{8}{27})^{1-x} =\frac{2}{3}\\((\frac{2}{3})^2)^x\cdot ((\frac{2}{3})^3)^{1-x} =\frac{2}{3}\\(\frac{2}{3})^{2x}\cdot (\frac{2}{3})^{3-3x} =\frac{2}{3}\\(\frac{2}{3})^{2x+3-3x} =\frac{2}{3}\\(\frac{2}{3})^{-x+3} =\frac{2}{3}\\-x+3=1\\-x=1-3\\-x=-2\ /:(-1)\\x=2

exis [7]3 years ago
7 0
X=2

All this stuff is wrong I screwed up somewhere:
IGNORE EVERYTHING BELOW THIS:
take the natural log of both sides of the equation.

ln(( \frac{4}{9})^{x}  )   \times  \ ln(( \frac{8}{27})^{(1 - x)} ) =  ln( \frac{2}{3} )
Remembering your log rules:

ln( {a}^{b} )  = b  \: ln(a)
Using this rule we rewrite the first equation:
x \:  ln( \frac{4}{9} )  \times (1 - x) \:  ln( \frac{8}{27} )  =  ln( \frac{2}{3} )
Simplify:
x ln( \frac{4}{9} )  \times ( ln( \frac{8}{27}) - x \:  ln( \frac{8}{27} )  )
Solve for some of these ln's to make life easier:

- .81x \:  \times ( - 1.22 - 1.22x) =  - .41
Distribute:
(.988x + .988x^{2} ) =  - .41

Factor out .988
.988(x +  {x}^{2} ) =  - .41 \\ (x +  {x}^{2} ) =  \frac{ - .41}{.988}  \\ (x +  {x}^{2} ) = -  .415
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