SOLUTION: Hey can you solve this please?: log4/9[(2x+1)/(x+3)]=-1/2

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.

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