Calculating the probability of several random variables with nonlinear constraints. For example, I have three independent random variables $x$,$y$,$z$, all of them are uniformly distributed between $0$ and $1$. Now I want to find the probability satisfying the two constraints simultaneously:
$$P\left[\frac{\log\left(x\right)}{\log\left(y\right)}<\frac{4}{3},\frac{\log\left(y\right)}{\log\left(z\right)}<\frac{3}{2}\right]$$
I know the basic method to find the volume by integration manually, and I can obtain the answer $4/15$. However it is too complicated and  I could have more of these variables and more constraints. 
My question is, for such problem, is there a systematic way or formula to obtain the probability  directly? Or, can these be transformed into exponential distributions and simplify the calculation?
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
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The answer is given by:
\begin{align}
&\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}
\bracks{{\ln\pars{x} \over \ln\pars{y}} < {4 \over 3}}
\bracks{{\ln\pars{y} \over \ln\pars{z}} < {3 \over 2}}\dd x\,\dd y\,\dd z
\\[5mm] = &\
\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}
\bracks{x > y^{4/3}}\bracks{z < y^{2/3}}\dd x\,\dd z\,\dd y =
\int_{0}^{1}\int_{0}^{y^{\large{2/3}}}\int_{y^{\large{4/3}}}^{1}
\dd x\,\dd z\,\dd y
\\[5mm] = &\
\int_{0}^{1}y^{2/3}\pars{1 - y^{4/3}}\,\dd y =
\int_{0}^{1}\pars{y^{2/3} - y^{2}}\,\dd y =
\left.\pars{{y^{5/3} \over 5/3} - {y^{3} \over 3}}\right\vert_{\ 0}^{\ 1} =
{3 \over 5} - {1 \over 3} = \bbx{4 \over 15}
\end{align}
