1. The **product rule** states that $\log_b(xy) = \log_b x + \log_b y$. This means the log of a product is the sum of the logs. 2. The **quotient rule** states that $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means the log of a quotient is the difference of the logs. 3. The **power rule** states that $\log_b(x^k) = k \log_b x$. This means the log of a power is the exponent times the log. 4. The **change of base formula** is $\log_b x = \frac{\log_a x}{\log_a b}$ where $a$ is any positive base different from 1. This allows changing the base of logarithm. 5. The **logarithm of 1** is always zero: $\log_b 1 = 0$, because $b^0 = 1$. 6. The **logarithm of the base** is always one: $\log_b b = 1$, because $b^1 = b$. These rules apply for any base $b > 0$, $b \neq 1$, and arguments $x, y > 0$.