[Example 5.4.3] If $2<a<4, b=(\sqrt{2})^{a}, c=\log _{\sqrt{2}} \log _{\sqrt{2}} b$, try to compare the sizes of $a, b$, and $c$.


Given the range of $a$: $2<a<4$, we now examine the ranges of $b$ and $c$.
$$
\begin{array}{c}
(\sqrt{2})^{2}<b=(\sqrt{2})^{a}<(\sqrt{2})^{4}, \\
2<b<4 .
\end{array}
$$

Also, $\log _{\sqrt{2}} 2<\log _{\sqrt{2}} b<\log _{\sqrt{2}} 4$,
$$
\begin{array}{c}
2<\log _{\sqrt{2}} b<4, \\
2<\log _{\sqrt{2}} \log _{\sqrt{2}} b<4, \\
2<c<4 .
\end{array}
$$

Thus, $a, b, c$ are all within the interval $(2,4)$, making it difficult to compare their sizes.
However, from $a=\log _{\sqrt{2}} b$ we know $c=\log _{\sqrt{2}} a$,
and since $b=(\sqrt{2})^{a}$,
$b$ and $c$ are inverse functions of each other.
Therefore, we can plot the graphs of
$$
y=x, y=(\sqrt{2})^{x}, y=\log _{\sqrt{2}} x
$$

From the graphs, we can see that
$$
b<a<c .
$$