# Log Functions Inside Absolute Value

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!

This function is equivalent to $$\mathrm{artanh}\,(\ln{(x)})$$ which is only defined for $$\frac1e\lt x\lt e$$ and is negative for $$\frac1e\lt x\lt1$$.
Note that excluding $$x=1/e$$ (for which your fraction does not exist), we have in the interval $$0 that $$log (x)<0$$, so the fraction will evaluate to something with absolute value smaller than 1, so the complete expression will be negative.