Question

How do I solve for x:

(4/9)^(x) • (8/27)^(1-x) =2/3

Answer 1

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(( (4)/(9))^(x) ) * \ ln(( (8)/(27))^((1 - x)) ) = ln( (2)/(3) )`
Remembering your log rules:


`ln( {a}^(b) ) = b \: ln(a)`
Using this rule we rewrite the first equation:

`x \: ln( (4)/(9) ) * (1 - x) \: ln( (8)/(27) ) = ln( (2)/(3) )`
Simplify:

`x ln( (4)/(9) ) * ( ln( (8)/(27)) - x \: ln( (8)/(27) ) )`
Solve for some of these ln's to make life easier:


`- .81x \: * ( - 1.22 - 1.22x) = - .41`
Distribute:

`(.988x + .988x^(2) ) = - .41`

Factor out .988

`.988(x + {x}^(2) ) = - .41 \\ (x + {x}^(2) ) = ( - .41)/(.988) \\ (x + {x}^(2) ) = - .415`

Answer 2

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