$$ \log x = \log_{10} x $$
$$ y = \log x \ \Longleftrightarrow \ x = 10^y $$
$$
\begin{aligned}
\log x &= \frac{\log_a x}{\log_a 10} = \frac{\ln x}{\ln 10} \\ \\
\log x &= \frac{1}{\log_x 10}
\end{aligned}
$$
$$
\begin{aligned}
& \log\left(x\cdot z\right) = \log x + \log z \\ \\
& \log\left(\frac{x}{z}\right) = \log x - \log z
\end{aligned}
$$
$$
\begin{aligned}
\log 1 &= 0 \\ \\
\log 10 &= 1
\end{aligned}
$$
$$
\begin{aligned}
& \log x^r = r\cdot\log x \\ \\
& \log 10^r = r \\ \\
& 10^{\log x} = \log 10^x = x
\end{aligned}
$$
