What is the exact form and corrected to three decimal places if not an integer of ln (x^2-1)-ln(x-1)=ln6
Solve for x:
log(x^2 - 1) - log(x - 1) = log(6)
log(x^2 - 1) - log(x - 1) = log(1/(x - 1)) + log(x^2 - 1) = log((x^2 - 1)/(x - 1)):
log((x^2 - 1)/(x - 1)) = log(6)
Cancel logarithms by taking exp of both sides:
(x^2 - 1)/(x - 1) = 6
Multiply both sides by x - 1:
x^2 - 1 = 6 (x - 1)
Expand out terms of the right hand side:
x^2 - 1 = 6 x - 6
Subtract 6 x - 6 from both sides:
x^2 - 6 x + 5 = 0
The left hand side factors into a product with two terms:
(x - 5) (x - 1) = 0
Split into two equations:
x - 5 = 0 or x - 1 = 0
Add 5 to both sides:
x = 5 or x - 1 = 0
Add 1 to both sides:
x = 5 or x = 1
log(x^2 - 1) - log(x - 1) ⇒ log(1^2 - 1) - log(1 - 1) = (undefined)
log(6) ⇒ log(6) ≈ 1.79176:
So this solution is incorrect
log(x^2 - 1) - log(x - 1) ⇒ log(5^2 - 1) - log(5 - 1) = log(2) + log(3) ≈ 1.79176
log(6) ⇒ log(6) ≈ 1.79176:
So this solution is correct
The solution is:
Answer: |x = 5
ln (x^2-1) - ln(x-1) = ln(6) write as
ln [ (x^2 - 1) / (x - 1) ] = ln 6 we can forget the logs
(x^2 - 1) / (x - 1) = 6 rearrange
x^2 - 1 = 6 (x - 1) simplify
x^2 - 1 = 6x - 6
x^2 - 6x + 5 = 0 factor
(x - 1) ( x -5) = 0 set each factor to 0 and x = 1 [reject because it makes one or more logs undefined] or x = 5 which is valid..