3. If $\log _{2}\left[\log _{\frac{1}{2}}\left(\log _{2} x\right)\right]=\log _{3}\left[\log _{\frac{1}{3}}\left(\log _{3} y\right)\right]=$ $\log _{5}\left[\log _{\frac{1}{5}}\left(\log _{5} z\right)\right]=0$, then
(A) $z<x<y$;
(B) $x<y<z$;
(C) $y<z<x$;
(D) $z<y<x$.
3. (A)

By the properties of logarithms, we have
$$
x=2^{\frac{1}{2}}, \quad y=3^{\frac{1}{3}}, \quad z=5^{\frac{1}{5}} .
$$

Using the monotonicity of power functions, compare the sizes of $x, y, z$.
$$
\begin{array}{lc}
\text { By } & x^{6}=\left(2^{\frac{1}{2}}\right)^{6}=2^{3}=8, \\
& y^{6}=\left(3^{\frac{1}{3}}\right)^{6}=3^{2}=9 . \\
\therefore & x<y . \\
\text { By } & x^{10}=\left(2^{\frac{1}{2}}\right)^{10}=2^{5}=32, \\
& z^{10}=\left(5^{\frac{1}{5}}\right)^{10}=5^{2}=25, \\
\therefore & z<x . \\
\text { Therefore } & z<x<y .
\end{array}
$$

Therefore