Question 890826: Hey can you solve this please?: log4/9[(2x+1)/(x+3)]=-1/2
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if i understand this correctly, the problem is:
log4/9(2x+1)/(x+3)=-1/2
this is true if and only if (4/9)^(-1/2) = (2x+1)/(x+3)
(4/9)^(-1/2) = 1/(4/9)^(1/2) = 1/(2/3) = 3/2
equation becomes:
3/2 = (2x+1)/(x+3)
multiply both sides of the equation by 2 and multiply both sides of the equation by (x+3) to get:
3(x+3) = 2(2x+1)
simplify to get:
3x+9 = 4x+2
subtract 3x from both sides of the equation and subtract 2 from both sides of the equation to get:
7 = x
that's your solution.
x = 7
when x = 7, your equation of log4/9[(2x+1)/(x+3)]=-1/2 becomes:
log4/9((2*7+1)/(7+3)) = -1/2 which becomes:
log4/9((15/10)) = -1/2 which becomes:
log4/9((3/2) = -1/2
you can solve this by translating it into an exponential equation.
you will get:
log4/9((3/2) = -1/2 if and only if (4/9)^(-1/2) = 3/2
(4/9)^(-1/2) is equal to 1 / (4/9)^(1/2.
you get:
1/(4/9)^(1/2) = 3/2 which becomes:
1/(2/3) = 3/2 which becomes:
3/2 = 3/2, confirming that x = 7 is the solution.
you could also have solved it by using the log base conversion formula and then using your calculator to get the log.
start with:
log4/9((3/2) = -1/2
log conversion formula says that log4/9(3/2) = LOG(3/2)/LOG(4/9)
LOG is equal to log10 which is the same as log to the base of 10 which is what your calculator can handle.
your equation becomes:
LOG(3/2)/LOG(4/9) = -1/2
use your calculator to get:
LOG(3/2)/LOG(4/9) = -.5.
-.5 is the same as -1/2 so you're good and the solution is x = 7.
|
|
|
