Here is one way of looking at it. (I'll assume that the numbers $a,b,x \in \mathbb R$ satisfy $a > 1, b > 1$, and $x > 0$.)
I dislike the name "logarithm" and I think a more descriptive name for $\log_b(x)$ is "the exponent from $b$ to $x$".
We could also use the notation $[b \to x]$ instead of $\log_b(x)$.
The change of base rule then tells us that the exponent from $b$ to $x$ is equal to the exponent from $b$ to $a$ times the exponent from $a$ to $x$:
$$
\tag{$\spadesuit$}[b \to x] = [b \to a][a \to x]
$$
or equivalently
$$
[a \to x] = [b \to x]/[b \to a].
$$
In standard notation, this formula states that
$$
\log_a(x) = \frac{\log_b(x)}{\log_b(a)}.
$$
Note that equation $(\spadesuit)$ is obvious, because
\begin{align}
b^{[b \to a][a \to x]} &=(b^{[b\to a]})^{[a \to x]} \\
&= a^{[a \to x]} \\
&= x.
\end{align}
Not an answer "per se" but two illuminating references (see below) about 17th century questionning on the connection between geometry and analysis.
These documents address transitions in what could be called the "status" of logarithms. Considered as practical mysterious "numbers" (the vision given by their discoverer, Napier, early 17th century), this status has only gradually moved into our modern vision of "log" as a "function". One of the key discoveries has been that these "logarithms" could be defined as areas under hyperbola $y=1/x$ ; many interesting "disputatios" about the mechanical / non-mechanical nature of this curve, the constructibility of certain of its "values" (the subject of this question) etc. have taken place at that time.
All these debates have allowed to build the modern vision of "log" , established on solid grounds by Euler (mid-18th century)
a) A document about "Descartes Logarithm machine" that can be found there :
http://www.quadrivium.info/MathInt/MathIntentions.html
b) A whole book available online :
"Impossibility results : from Geometry to Analysis"
https://hal.archives-ouvertes.fr/tel-01098493/document
by Davide Crippa, an historian of science. A very interesting document, devoting in particular a large place to the huge work and genial findings of Huygens, that has paved the way to the discovery of calculus (about logarithms, see in particular pages 446-450).
Best Answer
$\log_b(k)$ is defined as the power that $b$ needs to be raised to produce $k$. Clearly, if $b$ is raised to this power, then the result will produce $k$. Hence, $b^{\log_b(k)}=k$.