1. The logarithm is the inverse operation of exponentiation. 2. It answers the question: To what power must the base $b$ be raised, to produce the number $x$? 3. Mathematically, if $b^y = x$, then $\log_b(x) = y$. 4. Here, $b$ is the base, $x$ is the argument, and $y$ is the logarithm. 5. The base $b$ must be positive and not equal to 1 ($b>0$, $b \neq 1$). 6. For example, $\log_2(8) = 3$ because $2^3 = 8$. 7. Common bases are 10 (common logarithm) and $e$ (natural logarithm, denoted $\ln$). 8. Key properties include: - $\log_b(xy) = \log_b(x) + \log_b(y)$ - $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$ - $\log_b(x^k) = k\log_b(x)$ 9. These rules help simplify and solve exponential and logarithmic equations.