From the semiotically active logarithm notation (abbreviated Mirabilis) we obtain the properties that are used in current times. Next, we present the properties.

Law of Multiplication (Multiplicatio Canon)

$$ \begin{aligned} \displaystyle \log_{2^6} 64 &= \\ &= \log_{2^6} 4~.~16 = \log_{2^2} 4 + \log_{2^4} 16 \\ &= \log_{2^6} 8~.~8 = \log_{2^3} 4 + \log_{2^3} 16 \\ &= \log_{2^6} 2~.~32 = \log_{2^1} 2 + \log_{2^5} 32 \\ \end{aligned} $$
$$\log_{C} A~.~B = \log_{C} A + \log_{C} B$$

Law of division (Tangentiplus Canon)

$$ \begin{aligned} \log_{2^4} \frac{64}{4} &= \log_{2^6} 64 - \log_{2^2} 4 \\ \log_{5^1} \frac{125}{25} &= \log_{5^3} 125 - \log_{5^2} 25 \end{aligned} $$
$$\log_{C} \frac{A}{B} = \log_{C} A - \log_{C} B$$

Law of Power (Potestati Canon)

$$ \begin{aligned} \log_{2^6} 4^3 &= \log_{2^6} 64\\ &= \log_{2^6} 4 ~.~ 4 ~.~ 4\\ &\overset{\text{Multiplicatio Canon}}{=} \log_{2^2} 4 + \log_{2^2} 4 + \log_{2^2} 4\\ &= 3~.~\log_{2^2} 4\\ \end{aligned} $$
$$\log_{C} A^B = B~.~\log_{C} A$$

Application for Planck constant

To understand the power of this notation, let’s calculate Planck’s constant.

$$\hbar = \frac{ 81^{\frac{1}{3.1}} ~.~ 27^{\frac{1}{2.8}} }{ 243^{\frac{1}{10.2}} }$$
$$ \begin{aligned} \log \hbar &= \log \frac{ 81^{\frac{1}{3.1}} ~.~ 27^{\frac{1}{2.8}} }{ 243^{\frac{1}{10.2}} }\\ &= \log 81^{\frac{1}{3.1}} + \log 27^{\frac{1}{2.8}} - \log 243^{\frac{1}{10.2}} \\ &= \frac{1}{3.1}\log 81 + \frac{1}{2.8}\log 27 - \frac{1}{10.2}\log 243 \\ &= \frac{1}{3.1}\log_3 81 + \frac{1}{2.8}\log_3 27 - \frac{1}{10.2}\log_3 243 \\ &= \frac{1}{3.1}\log_{3^4} 81 + \frac{1}{2.8}\log_{3^3} 27 - \frac{1}{10.2}\log_{3^5} 243 \\ &= \frac{4}{3.1} + \frac{3}{2.8} - \frac{5}{10.2} \\ &= 1.3 + 1.2 - 0.5 = 2 \\ \end{aligned} $$
$$\log_{3^2} \hbar \qquad \Rightarrow\qquad \hbar = 9$$

Application for similarity of triangles

This application was known as Kanon trigon sunomols or Canon angulus sinonimus, more specifically Law of similarity of triangles.

$$\frac{3}{4} = \frac{6}{8}$$

In fact, on the left side applying the log we have:

$$\log\frac{3}{4} \overset{\text{tangentiplus canon}}{=} \log 3 - \log 4$$

On the other hand,

$$ \begin{aligned} \log \frac{6}{8} &= \log 6 - \log 8\\ &\overset{\text{multiplicatio canon}}{=} \log 2~.~3 - \log 2~.~4\\ &= \color{red}{\log 2} + \log 3 \color{red}{-\log 2} - \log 4\\ &= \log 3 - \log 4 = \log\frac{3}{4} \end{aligned} $$

In this way, the similarity property of triangles holds.