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

[jenny_]

Hi,
I'm hitting the wall to solve this problem as title, show that -ln(x-sqrt(x^2-1)) = ln(x+sqrt(x^2-1)). This is problem from my precalculus text book by jame stewart in logarithm function section.

I really have no idea why they are identical graph. I can clearly see that they are same in graph calculator and put arbitrary number in it, getting same number. But why? Can anyone help me understand what's the rationale behind it?

 

[stapel]

Hi,
I'm hitting the wall to solve this problem as title, show that -ln(x-sqrt(x^2-1)) = ln(x+sqrt(x^2-1)).

When you're not sure, just try stuff. In this case:

Apply the log rule that takes multipliers out front (the [imath]-1[/imath] in this case) inside as exponents. Apply that exponent to the argument.

You now have a fraction inside the log, and that fraction contains a radical in its denominator. So rationalize the denominator. What do you get?

 

[Steven G]

Major hint: 1/(x-sqrt(x^2-1) = x+sqrt(x^2-1)
How did I get that that equality? Why is that equal sign valid?