1. Let's start by stating the problem: You want to understand the rules of logarithms. 2. The logarithm of a number answers the question: to what power must we raise a base number to get that number? The general form is $\log_b(x) = y$ means $b^y = x$ where $b$ is the base, $x$ is the argument, and $y$ is the logarithm. 3. Important rules of logarithms are: - 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 $a \neq 1$ 4. These rules come from the properties of exponents. For example, the product rule comes from $b^m \cdot b^n = b^{m+n}$, so taking logs turns multiplication into addition. 5. Let's see an example using the product rule: Calculate $\log_2(8 \times 4)$. Using the product rule: $$\log_2(8 \times 4) = \log_2(8) + \log_2(4)$$ Since $8 = 2^3$ and $4 = 2^2$, we have: $$\log_2(8) = 3, \quad \log_2(4) = 2$$ So: $$\log_2(8 \times 4) = 3 + 2 = 5$$ 6. This means $2^5 = 32$, and indeed $8 \times 4 = 32$. 7. Understanding these rules helps simplify logarithmic expressions and solve logarithmic equations easily.