The students are taught the well known change base rule of logarithm:
\begin{align}\log_a b = \frac{\log_c b}{\log_c a}\end{align} Most text books proves it by invoking $(a^x)^y=a^{xy}$ to show:
\begin{align}\log_c a\times\log_a b=\log_c b\end{align}
Question:
Why don't we call it Chain Rule of Logarithm (at least as a second name)?
Is it because the way we use it is almost alway in the change of base format or something else?
\begin{align}
\log_a b\times\log_b c\times\log_c d\times\log_d e\times\cdots\times\log_y z=\log_a z\end{align}
It is easy to remember and the students can have fun to continue chaining it (similar to the change rule of derivative).
Best Answer
A side note:
The argument of the current factor must be equal to the base of following factor. I´ve colored the corresponding parameters.
$$\begin{align}\log_{\color{blue}{a}} \color{red}{b}\times\log_{\color{red}{b}} \color{orange}{c}\times\log_{\color{orange}{c}} \color{green}{d}\times\cdots\times\log_y \color{yellowgreen}{z}=\log_{\color{blue}{a}} \color{yellowgreen}z\end{align}$$
I hope you see the difference to your term. The chain rule is about derivatives and the concept is very different from the rule you´ve posted.
It is more related to the overall growth rate $r$, if you have n consecutive growth rates ($r_i$), with $r_i=\frac{y_{i+1}}{y_i}-1$.
$$1+r=\frac{y_{1}}{y_0}\cdot \frac{y_{2}}{y_1}\cdot \ldots \cdot \frac{y_{n-1}}{y_{n-2}} \cdot \frac{y_{n}}{y_{n-1}}=\frac{y_{n}}{y_{0}}$$
The rule you´ve posted is related to the concept of the geometric mean.