1. Problema: Desarrollar aplicando las propiedades de los logaritmos las siguientes expresiones. 2. Propiedades importantes de logaritmos: - $\log_b(xy) = \log_b x + \log_b y$ - $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$ - $\log_b(x^n) = n \log_b x$ - $\ln$ es logaritmo natural (base $e$). 3. Desarrollo: i. $$\ln \left[ x^3 (x+2) \right]^4 = 4 \ln \left( x^3 (x+2) \right) = 4 \left( \ln x^3 + \ln (x+2) \right) = 4 \left( 3 \ln x + \ln (x+2) \right) = 12 \ln x + 4 \ln (x+2)$$ ii. $$\log_2 \left[ (x+1)^3 (x^2 - 3) x^5 \right] = \log_2 (x+1)^3 + \log_2 (x^2 - 3) + \log_2 x^5 = 3 \log_2 (x+1) + \log_2 (x^2 - 3) + 5 \log_2 x$$ iii. $$\log \sqrt{x (x+4) (x-3)^2} = \log \left[ x (x+4) (x-3)^2 \right]^{1/2} = \frac{1}{2} \log \left[ x (x+4) (x-3)^2 \right] = \frac{1}{2} \left( \log x + \log (x+4) + \log (x-3)^2 \right) = \frac{1}{2} \left( \log x + \log (x+4) + 2 \log (x-3) \right) = \frac{1}{2} \log x + \frac{1}{2} \log (x+4) + \log (x-3)$$ iv. $$\ln \left[ \frac{x^5}{x+4} \right] = \ln x^5 - \ln (x+4) = 5 \ln x - \ln (x+4)$$ v. $$\log \left[ \frac{x^5}{(x+2)(x-3)} \right] = \log x^5 - \log (x+2) - \log (x-3) = 5 \log x - \log (x+2) - \log (x-3)$$ vi. $$\log_5 \left[ \frac{x-7}{x+2} \right]^4 = 4 \log_5 \left( \frac{x-7}{x+2} \right) = 4 \left( \log_5 (x-7) - \log_5 (x+2) \right) = 4 \log_5 (x-7) - 4 \log_5 (x+2)$$ vii. $$\log_3 \left[ \frac{(x+4)^2 (x-3)}{\sqrt{x}} \right] = \log_3 (x+4)^2 + \log_3 (x-3) - \log_3 x^{1/2} = 2 \log_3 (x+4) + \log_3 (x-3) - \frac{1}{2} \log_3 x$$ viii. $$\ln \left[ \frac{x^3 \sqrt{x} + 5}{x+3} \right]$$ No se puede simplificar más sin información adicional, queda expresado así. ix. $$\log \sqrt{ \frac{x^2 (x+2)}{(x+8)^4 (x-3)^6} } = \log \left[ \frac{x^2 (x+2)}{(x+8)^4 (x-3)^6} \right]^{1/2} = \frac{1}{2} \log \left[ \frac{x^2 (x+2)}{(x+8)^4 (x-3)^6} \right] = \frac{1}{2} \left( \log x^2 + \log (x+2) - \log (x+8)^4 - \log (x-3)^6 \right) = \frac{1}{2} \left( 2 \log x + \log (x+2) - 4 \log (x+8) - 6 \log (x-3) \right) = \log x + \frac{1}{2} \log (x+2) - 2 \log (x+8) - 3 \log (x-3)$$ 4. Resumen final: i. $12 \ln x + 4 \ln (x+2)$ ii. $3 \log_2 (x+1) + \log_2 (x^2 - 3) + 5 \log_2 x$ iii. $\frac{1}{2} \log x + \frac{1}{2} \log (x+4) + \log (x-3)$ iv. $5 \ln x - \ln (x+4)$ v. $5 \log x - \log (x+2) - \log (x-3)$ vi. $4 \log_5 (x-7) - 4 \log_5 (x+2)$ vii. $2 \log_3 (x+4) + \log_3 (x-3) - \frac{1}{2} \log_3 x$ viii. $\ln \left[ \frac{x^3 \sqrt{x} + 5}{x+3} \right]$ ix. $\log x + \frac{1}{2} \log (x+2) - 2 \log (x+8) - 3 \log (x-3)$