1. Let's start by understanding what a logarithm is. A logarithm answers the question: to what power must we raise a base number to get another number? For example, $\log_b(a) = c$ means $b^c = a$. 2. The general formula for logarithms is: $$\log_b(a) = c \iff b^c = a$$ where $b$ is the base, $a$ is the result, and $c$ is the exponent. 3. Important rules of logarithms include: - 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: $\log_b(x^k) = k \log_b(x)$ 4. Let's do an example: Find $\log_2(8)$. Since $2^3 = 8$, we have $\log_2(8) = 3$. 5. Another example: Simplify $\log_3(27) + \log_3(9)$. Since $27 = 3^3$ and $9 = 3^2$, by the product rule: $$\log_3(27) + \log_3(9) = \log_3(27 \times 9) = \log_3(243)$$ And $243 = 3^5$, so: $$\log_3(243) = 5$$ 6. Logarithms are very useful in solving equations where the unknown is an exponent. In summary, logarithms convert multiplication into addition, division into subtraction, and powers into multiplication, making complex calculations easier.