1. **State the problem:** Solve the logarithmic equation or expression given (user did not specify the exact problem, so we will explain general solving steps for logarithmic equations). 2. **Recall the logarithm definition and properties:** - The logarithm $\log_b(a)$ answers the question: "To what power must we raise $b$ to get $a$?" - Key properties: - $\log_b(xy) = \log_b(x) + \log_b(y)$ - $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$ - $\log_b(x^r) = r \log_b(x)$ - Change of base formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ 3. **General approach to solve logarithmic equations:** - Isolate the logarithmic expression if possible. - Use properties to combine or simplify logs. - Convert the logarithmic equation to exponential form: if $\log_b(x) = y$, then $x = b^y$. - Solve the resulting algebraic equation. - Check for extraneous solutions (logarithm arguments must be positive). 4. **Example:** Solve $\log_2(x) + \log_2(x-3) = 3$ - Use product property: $\log_2(x) + \log_2(x-3) = \log_2(x(x-3))$ - So, $\log_2(x(x-3)) = 3$ - Convert to exponential form: $x(x-3) = 2^3$ - Simplify: $x^2 - 3x = 8$ - Rearrange: $x^2 - 3x - 8 = 0$ - Solve quadratic: $x = \frac{3 \pm \sqrt{9 + 32}}{2} = \frac{3 \pm \sqrt{41}}{2}$ - Approximate roots: $x \approx 5.7$ or $x \approx -1.7$ - Check domain: $x > 0$ and $x-3 > 0 \Rightarrow x > 3$ - Only $x \approx 5.7$ is valid. 5. **Final answer:** $x = \frac{3 + \sqrt{41}}{2}$