1. Let's start by stating the problem: We want to understand logarithms better, including their properties and how to solve logarithmic equations. 2. The logarithm is the inverse operation of exponentiation. The logarithm base $b$ of a number $x$ is written as $\log_b(x)$ and means the exponent to which $b$ must be raised to get $x$. Formally, if $b^y = x$, then $\log_b(x) = y$. 3. Important properties 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: $\log_b(x^k) = k \log_b(x)$ - Change of base formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ for any positive $a \neq 1$ 4. Let's solve an example: Solve for $x$ in $\log_2(x) + \log_2(x-3) = 3$. 5. Using the product rule, combine the logs: $\log_2(x(x-3)) = 3$. 6. This means $2^3 = x(x-3)$, so $8 = x^2 - 3x$. 7. Rearranging: $x^2 - 3x - 8 = 0$. 8. Solve the quadratic equation using the quadratic formula: $x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 + 32}}{2} = \frac{3 \pm \sqrt{41}}{2}$. 9. Calculate the roots: $x = \frac{3 + \sqrt{41}}{2} \approx 5.7$ and $x = \frac{3 - \sqrt{41}}{2} \approx -1.7$. 10. Since logarithms require positive arguments, discard $x \approx -1.7$ because $x-3$ would be negative. 11. Final answer: $x \approx 5.7$. This example shows how to apply logarithm properties and solve equations step-by-step.