As I understand this logarithmic functions are a family of functions where the equation for $f(x)$ is written like so
$$\begin{align} f(x) = & \log_{a} x \\ & \mathtt{where\ }a\mathtt{\ is\ the\ independent\ variable} \end{align} $$
Now I'm curious about general applications of this kind of functions in algebra and geometry, if I'm not mistaking they have some nice properties regarding proportions and they are useful to build shapes ? I don't know I'm quite confused on this one; Some can explain why this kind of functions are popular ?
$\mathtt{Edit :}$
this is an interesting little compact explanation for example, but it lacks a real explanation and some examples .
It is not totally clear what you are asking, but one nice property is that if $a$ and $b$ are positive but not $1$ then $$\log_a x = k \log_b x$$ for some constant $k$ and all positive $x$. So logarithms to any base are scaled versions of logarithms to other bases.
In fact $k=\log_a b = \dfrac{1}{\log_b a} = \dfrac{\log_c b}{\log_c a}$.