1. **State the problem:** Simplify and understand the logarithmic expressions and their derivatives using the laws of logarithms. 2. **Recall the laws of logarithms:** - $\ln(xy) = \ln x + \ln y$ - $\ln\left(\frac{x}{y}\right) = \ln x - \ln y$ - $\ln x^n = n \ln x$ - $\ln e^x = x$ - $e^{\ln x} = x$ 3. **Examples:** **Example 1:** Simplify $\ln \frac{x^2 - 9}{(x(x+4))^2}$ - Use quotient rule: $\ln \frac{A}{B} = \ln A - \ln B$ - $= \ln (x^2 - 9) - \ln (x^2 (x+4)^2)$ - Factor $x^2 - 9 = (x-3)(x+3)$ - $= \ln (x-3) + \ln (x+3) - \ln x^2 - \ln (x+4)^2$ - Use power rule: $\ln x^2 = 2 \ln x$, $\ln (x+4)^2 = 2 \ln (x+4)$ - Final: $\ln (x-3) + \ln (x+3) - 2 \ln x - 2 \ln (x+4)$ **Example 2:** Simplify $\ln \sqrt{a^2 - x^2}$ - Rewrite square root as power: $\sqrt{a^2 - x^2} = (a^2 - x^2)^{1/2}$ - Use power rule: $\ln (a^2 - x^2)^{1/2} = \frac{1}{2} \ln (a^2 - x^2)$ **Example 3:** Simplify $\log \frac{1 + x^2}{1 - x^2}$ - Use quotient rule: $\log (1 + x^2) - \log (1 - x^2)$ **Example 4:** Simplify $\log_a (x)(x - 1)$ - This is $\log_a (x) + \log_a (x - 1)$ if multiplication inside log, but here it looks like product of logs? - If expression is $\log_a (x) (x - 1)$, it is $\log_a (x)$ times $(x-1)$, no simplification unless specified. 4. **Derivative of logarithms:** - $\frac{d}{dx} \log_a u = \frac{\frac{du}{dx}}{u \ln a}$ - $\frac{d}{dx} \ln u = \frac{\frac{du}{dx}}{u}$ since $\ln e = 1$ - $\frac{d}{dx} \log_{10} u = M \frac{\frac{du}{dx}}{u}$ where $M = \log_{10} e = 0.434 / 2.9$ **Summary:** - Use logarithm laws to simplify expressions by converting products to sums, quotients to differences, and powers to multipliers. - Derivatives of logarithms depend on the base and the chain rule applied to the inner function $u$.