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) $z<y<x$
Answer(
(A)
3.【Analysis and Solution】By the properties of logarithms, we have
$$
x=2^{\frac{1}{2}}, y=3^{\frac{1}{3}}, z=5^{\frac{1}{5}} \text {. }
$$

Using the monotonicity of power functions, compare the sizes of $x, y, z$.
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 $x^{10}=\left(2^{\frac{1}{2}}\right)^{10}=2^{5}=32$,
$$
\begin{array}{l}
z^{10}=\left(5^{\frac{1}{5}}\right)^{10}=5^{2}=25 . \\
\therefore z<x .
\end{array}
$$

Therefore, $z<x<y$.