Show that In x Squareroot x2 1 In x Squareroot x2 1Solu

Show that In (x - Squareroot x^2 - 1) = -In (x + Squareroot x^2 - 1)

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

Starting with the right side

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

and - ln a=ln a^-1

Therefore

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

rationalising the denominator

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

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