1. Let's start by understanding the logarithm (log). The logarithm answers the question: "To what power must we raise a certain base to get a number?" 2. The logarithm with base $b$ of a number $x$ is written as $\log_b(x)$ and means the exponent $y$ such that $b^y = x$. 3. For example, $\log_2(8) = 3$ because $2^3 = 8$. 4. Common logarithms use base 10 and are written as $\log(x)$. 5. Natural logarithms use base $e \approx 2.718$ and are written as $\ln(x)$. 6. Logarithm properties: - Product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$ - Quotient rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$ - Power rule: $\log_b(x^k) = k \log_b(x)$ - Change of base formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ for any positive $a \neq 1$ 7. Understanding these rules helps solve logarithmic equations, simplify expressions, and convert between exponential and logarithmic forms.