$$
\begin{aligned}
& e ≐ 2,718282 \\ \\
& \ln x = \log_{e} x
\end{aligned}
$$
$$ y = \ln x \ \Longleftrightarrow \ x = e^y $$
$$
\begin{aligned}
& \ln x = \frac{\log_a x}{\log_a e} = \frac{\log x}{\log e} \\ \\
& \ln x = \frac{1}{\log_x e}
\end{aligned}
$$
$$
\begin{aligned}
& \ln\left(x\cdot z\right) = \ln x + \ln z \\ \\
& \ln\left(\frac{x}{z}\right) = \ln x - \ln z
\end{aligned}
$$
$$
\begin{aligned}
& \ln 1 = 0 \\ \\
& \ln e = 1
\end{aligned}
$$
$$
\begin{aligned}
& \ln x^r = r\cdot\ln x \\ \\
& \ln e^r = r \\ \\
& e^{\ln x} = \ln e^x = x
\end{aligned}
$$
