Log Functions Inside Absolute Value

Question

Is the function below always positive for $0< x <1$? (I am determining if the function requires the modulus sign or not.)

$$\frac{1}{2}\log\left|\frac{1+\log(x)}{1-\log(x)}\right|$$

My first instinct is that it cannot be, but I would just like some third-party feedback.

Thank you!

Answer 1

This function is actually negative in your interval (plot it to verify).

Note that excluding $x=1/e$ (for which your fraction does not exist), we have in the interval $0<x<1$ that $log (x)<0$, so the fraction will evaluate to something with absolute value smaller than 1, so the complete expression will be negative.

Answer 2

This function is equivalent to $$\mathrm{artanh}\,(\ln{(x)})$$ which is only defined for $\frac1e\lt x\lt e$ and is negative for $\frac1e\lt x\lt1$.