Question

Ayuda por f

Con procedimento

Answer 1

Respuesta:

$=30$

Explicación paso a paso:

$\log _a\left(b^5\right)\log _c\left(a^3\right)\log _b\left(c^2\right)\\\\\mathrm{Simplificar}\:\log _a\left(b^5\right):\quad 5\log _a\left(b\right)\\\\=5\log _a\left(b\right)\log _c\left(a^3\right)\log _b\left(c^2\right)\\\\\mathrm{Simplificar}\:\log _c\left(a^3\right):\quad 3\log _c\left(a\right)\\\\=5\cdot \:3\log _a\left(b\right)\log _c\left(a\right)\log _b\left(c^2\right)\\\\\mathrm{Simplificar}\:\log _b\left(c^2\right):\quad 2\log _b\left(c\right)$

$=5\cdot \:3\cdot \:2\log _a\left(b\right)\log _c\left(a\right)\log _b\left(c\right)\\\\\mathrm{Multiplicar\:los\:numeros:}\:5\cdot \:3\cdot \:2=30\\\\=30\log _a\left(b\right)\log _c\left(a\right)\log _b\left(c\right)\\\\\mathrm{Aplicar\:las\:propiedades\:de\:los\:logaritmos}:\quad \log _a\left(b\right)=\frac{\ln \left(b\right)}{\ln \left(a\right)}\\\\\log _a\left(b\right)\log _c\left(a\right)=\frac{\ln \left(b\right)}{\ln \left(a\right)}\cdot \frac{\ln \left(a\right)}{\ln \left(c\right)}$

$=30\cdot \frac{\ln \left(b\right)}{\ln \left(a\right)}\cdot \frac{\ln \left(a\right)}{\ln \left(c\right)}\log _b\left(c\right)\\\\\mathrm{Simplificar}\\\\=30\cdot \frac{\ln \left(b\right)}{\ln \left(c\right)}\log _b\left(c\right)\\\\\mathrm{Multiplicar\:fracciones}:\quad \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c}\\\\=\frac{\ln \left(b\right)\cdot \:30\log _b\left(c\right)}{\ln \left(c\right)}$

$=\frac{30\cdot \frac{\ln \left(c\right)}{\ln \left(e\right)}}{\ln \left(c\right)}\\\\\mathrm{Multiplicar\:}30\cdot \frac{\ln \left(c\right)}{\ln \left(e\right)}\::\quad 30\ln \left(c\right)\\\\=\frac{30\ln \left(c\right)}{\ln \left(c\right)}\\\\\mathrm{Eliminar\:los\:terminos\:comunes:}\:\ln \left(c\right)\\\\=30$