$$ y = \log_a x \ \Longleftrightarrow \ x = a^y $$
$$
\begin{aligned}
& \log_a x = \frac{\log_b x}{\log_b a} = \frac{\log x}{\log a} = \frac{\ln x}{\ln a} \\ \\
& \log_a x = \frac{1}{\log_x a}
\end{aligned}
$$
$$
\begin{aligned}
& \log_a \left(x\cdot z\right) = \log_a x + \log_a z \\ \\
& \log_a \left(\frac{x}{z}\right) = \log_a x - \log_a z
\end{aligned}
$$
$$
\begin{aligned}
& \log_a 1 = 0 \\ \\
& \log_a a = 1
\end{aligned}
$$
$$
\begin{aligned}
& \log_a x^r = r\cdot\log_a x \\ \\
& \log_a a^r = r \\ \\
& a^{\log_a x} = \log_a a^x = x
\end{aligned}
$$
