## Task B-3.2.

If $\log _{2}\left[\log _{3}\left(\log _{4} x\right)\right]=\log _{3}\left[\log _{2}\left(\log _{4} y\right)\right]=\log _{4}\left[\log _{3}\left(\log _{2} z\right)\right]=0$, calculate $\log _{y}\left(\log _{z} x\right)$.
## Solution.

From $\log _{2}\left[\log _{3}\left(\log _{4} x\right)\right]=0$, by applying the definition of logarithm, we get:

$$
\begin{gathered}
\log _{3}\left(\log _{4} x\right)=1 \\
\log _{4} x=3 \\
x=4^{3}
\end{gathered}
$$

Similarly, from $\log _{3}\left[\log _{2}\left(\log _{4} y\right)\right]=0$, we get:

$$
\begin{gathered}
\log _{2}\left(\log _{4} y\right)=1 \\
\log _{4} y=2
\end{gathered}
$$

$$
y=4^{2}
$$

From $\log _{4}\left[\log _{3}\left(\log _{2} z\right)\right]=0$, it follows that:

$$
\begin{gathered}
\log _{3}\left(\log _{2} z\right)=1 \\
\log _{2} z=3 \\
z=2^{3}
\end{gathered}
$$

Now we have:

$$
\log _{y}\left(\log _{z} x\right)=\log _{16}\left(\log _{8} 64\right)=\log _{16} 2=\frac{1}{4}
$$

Note: Any accurately calculated value of $x, y, z$ is worth two points, and the remaining two are worth one point each.