algebra precalculus - Intuition behind logarithm change of base

I try to understand the actual intuition behind the logarithm properties and came across a post on this site that explains the multiplication and thereby also the division properties very nicely:

Suppose you have a table of powers of 2, which looks like this: (after revision)

$$\begin{array}{rrrrrrrrrr}
0&1&2&3&4&5&6&7&8&9&10\\
1&2&4&8&16&32&64&128&256&512&1024
\end{array}$$

Each column says how many twos you have to multiply to get the number in that column. For example, if you multiply 5 twos, you get $2\cdot2\cdot2\cdot2\cdot2=32$, which is the number in column 5.

Now suppose you want to multiply two numbers from the bottom row, say $16\cdot 64$. Well, the $16$ is the product of 4 twos, and the $64$ is the product of 6 twos, so when you multiply them together you get a product of 10 twos, which is $1024$.

I found that very helpful to understand the actual proofs for this property.

I still struggle to get the idea behind the change of base rule. I'm familiar with the proof that goes like:

$$\log_a x = y \implies a^y = x$$
$$\log_b a^y = \log_b x$$

$$y \cdot \log_b a = \log_b x$$
$$y = \frac{\log_b x}{\log_b a}$$

But can somehow provide a explanation in the style of the quoted answer why this actually works?