Multiple logarithmic integral

Question

How would one go to prove that

$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{\mathrm{d}(x, y,z)}{ \ln x + \ln y + \ln z} = - \frac{1}{2}$$

I'm not good handling multivariable integrals and nothing pops up in my head.

Note: It should be noted that for two variables or one variable the integral diverges.

Answer 1

Inspired by the denominator, which equals $\ln(xyz)$, we change variables, $$ u=xyz,\quad v=y,\quad w=z. $$ Then you will have the integral $$ \iiint_K\frac{1}{vw\ln u}\,du\,dv\,dw. $$ Here, $K$ is determined by the inequalities $$ \frac{u}{w}\leq v\leq 1,\quad u\leq w\leq 1,\quad 0\leq u\leq 1. $$ If you integrate in that given order (first $v$, then $w$ and last $u$), you will soon find that the integral equals $-1/2$. I leave those calculations to you.

Answer 2

Interview question finalized in one line

$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1}\frac{dx dydz }{ \log(x y z)}=\int_{0}^{1} \int_{0}^{1} \int_{0}^{1}\int_1^{\infty}-(xyz)^{s-1}dsdx dydz=\int_1^{\infty}-\frac{1}{s^3}ds=-\frac{1}{2}.$$