SOLUTION: Solve 3ln(y +4) = 2 ln(x +2) − 2ln(x +9)+ 3ln(x^2 − 1). i ) Determine the range of values of x and y for which the expressions on each side of this equation are d

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Solve 3ln(y +4) = 2 ln(x +2) − 2ln(x +9)+ 3ln(x^2 − 1). i ) Determine the range of values of x and y for which the expressions on each side of this equation are d      Log On


   

Question 401048: Solve
3ln(y +4) = 2 ln(x +2) − 2ln(x +9)+ 3ln(x^2 − 1).
i ) Determine the range of values of x and y for which the expressions on
each side of this equation are defined
ii ) Find y explicitly as a function of x, that is, express the equation in the
form y = f(x), simplifying your answer as far as possible.
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
Solve
3ln(y +4) = 2 ln(x +2) − 2ln(x +9)+ 3ln(x^2 − 1).
i ) Determine the range of values of x and y for which the expressions on
each side of this equation are defined
ii ) Find y explicitly as a function of x, that is, express the equation in the
form y = f(x), simplifying your answer as far as possible.
(i)For logx, x>0
3ln(y +4) = 2 ln(x +2) − 2ln(x +9)+ 3ln(x^2 − 1)
range & domain as follows:
y+4>0
y>-4 (-4,infinity)
x+2>0
x>-2 (-2,infinity)
x+9>0
x>-9 (not in domain)
x^2-1>0
x^2>1
x>1 or x<-1 (-infinity,-1)union(1,infinity)
Simplifying equation:
3ln(y +4) = 2 ln(x +2) − 2ln(x +9)+ 3ln(x^2 − 1)
ln(y+4)^3=ln(x+2)^2-ln(x+9)^2+ln(x^2-1)^3
0=ln(x+2)^2+ln(x^2-1)^3-ln(x+9)^2-ln(y+4)^3
0=ln(x+2)^2+ln(x^2-1)^3-(ln(x+9)^2+ln(y+4)^3)
0=ln((x+2)^2)*((x^2-1)^3)/((x+9)^2)*((y+4)^3)
e^0=(x+2)^2)*((x^2-1)^3)/((x+9)^2)*((y+4)^3)
1=(x+2)^2)*((x^2-1)^3)/((x+9)^2)*((y+4)^3)
(y+4)^3=(x+2)^2)*((x^2-1)^3)/((x+9)^2)
taking the cube root of each side
y+4=((x+2)^2/3)*(x^2-1)/((x+9)^2/3))
y=((x+2)^2/3)*(x^2-1)/((x+9)^2/3))-4
f(x)={((x+2)^2/3)*(x^2-1)/((x+9)^2/3)}-4