Show that log base b [(x + (sqrt x^2-5))/5] is equal to -log base b (x-(sqrt x^2-5))

Answer 1

(x+√(x^2-5))/5
multiply top and bottom by (x-√(x^2-5)) and you have

(x+√(x^2-5))(x-√(x^2-5))
---------------------------------
5(x-√(x^2-5))

since (a+b)(a-b) = a^2-b^2, that is

x^2-(x^2-5)
-----------------
5(x-√(x^2-5))

5
---------------
5(x-√(x^2-5))

1/(x-√(x^2-5))

since log(1/a) = -log(a), that answers your question.