SOLUTION: Find the exact value for the solution of the equation. log((x^2)+4)-log(x+2)=2+log(x-2) I have the answer in the back of my book and it says the answer is: (2/3)*(sqrt(101/11))=~

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Find the exact value for the solution of the equation. log((x^2)+4)-log(x+2)=2+log(x-2) I have the answer in the back of my book and it says the answer is: (2/3)*(sqrt(101/11))=~      Log On


   

Question 691108: Find the exact value for the solution of the equation.
log((x^2)+4)-log(x+2)=2+log(x-2)
I have the answer in the back of my book and it says the answer is: (2/3)*(sqrt(101/11))=~2.02
But every way I've tried didnt work. Please help! and thank you!
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
exact value for the solution of the equation.
log(x^2+4)-log(x+2) = 2+log(x-2)
subtracting indicates divide so we can write it
log%28%28%28x%5E2%2B4%29%2F%28x%2B2%29%29%29 = 2 + log(x-2)
Subtract log(x-2) from both sides
log%28%28%28x%5E2%2B4%29%2F%28x%2B2%29%29%29 - log(x-2) = 2
which we can combine
log%28%28%28x%5E2%2B4%29%2F%28%28x%2B2%29%28x-2%29%29%29%29 = 2
which is
log%28%28%28x%5E2%2B4%29%2F%28x%5E2-4%29%29%29 = 2
find the antilog (10^x) of both sides (10^2 = 100)
%28%28x%5E2%2B4%29%2F%28x%5E2-4%29%29 = 100
multiply both sides by x^2-4
x%5E2+%2B+4+=+100%28x%5E2-4%29
x%5E2+%2B+4+=+100x%5E2-400
combine like terms
0 = 100x^2 - x^2 - 400 - 4
99x^2 - 404 = 0
99x^2 = 404
x^2 = 404%2F99
x = sqrt%28404%2F99%29
x = sqrt%28%284%2A101%29%2F%289%2A11%29%29
Extract the square roots of the perfect squares
x =2%2F3*sqrt%28101%2F11%29
x ~ 2.02