Question 890826
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.