<s>[INST] How do you explain the concept of logarithm to a five year old? Okay, I understand that it cannot be explained to a 5 year old. But how do you explain the logarithm to primary school students?
 [/INST] A logarithm is how much bigger or smaller a number is than another
number when you combine numbers using multiplication instead
of addition.
For instance, to see how much bigger $3$ is than $2$ in this way:
$$
\begin{array}{|cccccccccccccccc|}
\hline
2^1 & \cdot & 2^2 & \color{red}{2^3} & \cdot & 2^4 & \cdot & 2^5 & 2^6 & \cdot & 2^7 & \cdot & \color{red}{2^8} & 2^9 & \cdot & 2^{10} \\
\hline
 2  & \cdot &  4  &  \color{red}{8}  & \cdot & 16  & \cdot & 32  & 64  & \cdot & 128 & \cdot & \color{red}{256} & 512 & \cdot & 1024   \\
\cdot & 3   & \cdot & \cdot &  \color{red}{9}  & \cdot & 27  & \cdot & \cdot & 81  & \cdot & \color{red}{243} & \cdot & \cdot & 729 & \cdot \\
\hline
\cdot & 3^1 & \cdot & \cdot & \color{red}{3^2} & \cdot & 3^3 & \cdot & \cdot & 3^4 & \cdot & \color{red}{3^5} & \cdot & \cdot & 3^6 & \cdot \\
\hline
\end{array}
$$
This shows that
$$
1.5 = \frac{\color{red}{3}}{\color{red}{2}} < \log_2{3} < \frac{\color{red}{8}}{\color{red}{5}} = 1.6.
$$
You can measure $\log_2{3}$ as accurately as you like by comparing bigger powers
of $2$ and $3$.
(After this, either you end up with one very confused five-year-old, 
or else you've got a budding Eudoxus on your hands.)
</s>