SOLUTION: a) write log(x^3 √(x+1)/(x-2)^2, x>2 as a sum and difference logarithms.Express all powers as factors. b) Find the exact value of the composite function cos[sin^-1(-1/3)]

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: a) write log(x^3 √(x+1)/(x-2)^2, x>2 as a sum and difference logarithms.Express all powers as factors. b) Find the exact value of the composite function cos[sin^-1(-1/3)]       Log On


   

Question 1189490: a) write log(x^3 √(x+1)/(x-2)^2, x>2 as a sum and difference logarithms.Express all powers as factors.
b) Find the exact value of the composite function cos[sin^-1(-1/3)]


Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
a) write log%28x%5E3+%2Asqrt%28%28x%2B1%29%29%2F%28x-2%29%5E2%29, x%3E2 as a sum and difference logarithms.
Express all powers as factors.

log%28x%5E3%2A+sqrt%28%28x%2B1%29%29%2F%28x-2%29%5E2%29

= log%28x%5E3%29%2Blog%28+sqrt%28%28x%2B1%29%29%2F%28x-2%29%5E2%29

= 3log%28x%29%2Blog%28+sqrt%28x%2B1%29%29-log%28%28x-2%29%5E2%29

= 3log%28x%29%2Blog%28%28x%2B1%29%5E%281%2F2%29%29-2log%28%28x-2%29%29

= 3log%28x%29%2B%281%2F2%29log%28%28x%2B1%29%29-2log%28%28x-2%29%29

True for all x.
Verify solution x%3E2: The solution is x%3E2

b) Find the exact value of the composite function cos%28sin%5E-1%28-1%2F3%29%29
cos%28sin%5E-1%28-1%2F3%29%29
let sin%5E-1%28-1%2F3%29=theta => sin%28theta%29=-1%2F3
This means that we are now looking for cos%28theta%29.
Next, use the identity :
cos%5E2%28theta%29%2Bsin%5E2%28theta%29=1
cos%28theta%29=sqrt%281-sin%5E2%28theta%29%29
cos%28theta%29=sqrt%281-%28-1%2F3%29%5E2%29
cos%28theta%29=sqrt%288%2F9%29
cos%28theta%29=%282sqrt%282%29%29%2F3

Answer by MathTherapy(10553) About Me  (Show Source):
You can put this solution on YOUR website!

a) write log(x^3 √(x+1)/(x-2)^2, x>2 as a sum and difference logarithms.Express all powers as factors.
b) Find the exact value of the composite function cos[sin^-1(-1/3)]
a) log+%28%28x%29%29%5E3+%2B+log%28%28sqrt%28x+%2B+1%29%29%29+-+log%28%28x+-+2%29%29%5E2
   log+%28%28x%29%29%5E3+%2B+log%28%28x+%2B+1%29%29%5E%281%2F2%29+-+log%28%28x+-+2%29%29%5E2 ------- Applying log-law: matrix%281%2C3%2C+sqrt%28a%29%2C+%22=%22%2C+a%5E%281%2F2%29%29