show that -ln(x-sqrt of x^2-1) = ln(x + sqrt of x^2 - 1)

Question

show that

-ln(x - sqrt (x^2 -1) = ln(x + sqrt (x^2 - 1).

Answer 1

-ln(x - sqrt(x^2 - 1)) = ln(x + sqrt(x^2 - 1))

ln( 1/ (x - sqrt(x^2 - 1)) ) = ln(x + sqrt(x^2 - 1))

1 / (x - sqrt(x^2 - 1)) = x + sqrt(x^2 - 1)

1 = (x + sqrt(x^2 - 1)) * (x - sqrt(x^2 - 1))

Note:  (a - b)(a + b) = a^2 - b^2

1 = x^2 - (sqrt(x^2 - 1))^2

1 = x^2 - (x^2 - 1)

1 = x^2 - x^2 + 1

1 = 1

True.