Root of Logarithmic Equation

Question

I'm studying a function:

$$f(x) = (x - 1) \log(x^2 - 1)$$

Having as first derivative:

$$f'(x) = \frac{(x+1)\log(x^2-1) + 2x}{ x + 1 }$$

I'm looking for critical points ($f'(x) = 0$). I know it has an approximate numerical solution (1.15) thanks to Wolfram Alpha, but I don't know how to compute it myself... I tried the usual steps for logarithmic equations (substitutions, properties of logarithms) but nothing seems to work.

Answer 1

You can only solve numerically. Other than Newton you can use the iteration $x_0=1$ and $$x_{n+1}=\sqrt{e^{-2x_n/(x_n+1)}+1}.$$ This formula comes from setting the numerator of $f'(x)$ to zero and applying the exponential function. The first iterates are:

1.00000
1.16956
1.15768
1.15843
1.15838
1.15838