Question

1)Desenvolva, aplicando as propriedades operatorias dos logaritmos( suponha a,b e c reais positivos):
a)log₅(5a/bc)
b)log(b²/10a)
c)log₃(ab²/c)
d)log₂(8a/b³c²)

Answer 1

Propriedades:

$\boxed{log_x\frac{a}{b} = log_{x}a-log_{x}b} \\\\ \boxed{log_{x}a \cdot b = log_{x}a+log_{x}b} \\\\ \boxed{log_{x}a^{b} = b \cdot log_{x} a}$


Vamos aos exercícios:

a) $log_{5}\frac{5a}{bc} \\\\ log_{5}5a-log_{5}bc \\\\ (log_{5}5+log_{5}a)-(log_{5}b+log_{5}c) \\\\ \boxed{\boxed{1+log_{5}a-log_{5}b-log_{5}c}}$


b) $log\frac{b^{2}}{10a} \\\\ logb^{2}-log10a \\\\ (2 \cdot logb)-(log10+loga) \\\\ 2logb - (1+loga) \\\\ \boxed{\boxed{2logb-1-loga}}$


c) $log_{3}\frac{ab^{2}}{c} \\\\ log_{3}ab^{2}-log_{3}c \\\\ (log_{3}a+log_{3}b^{2})-log_{3}c \\\\ (log_{3}a+2log_{3}b)-log_{3}c \\\\ \boxed{\boxed{log_{3}a+2log_{3}b-log_{3}c}}$


d) $log_{2}\frac{8a}{b^{3}c^{2}} \\\\ log_{2}8a-log_{2}b^{3}c^{2} \\\\ (log_{2}8+log_{2}a)-(log_{2}b^{3}+log_{2}c^{2}) \\\\ (3+log_{2}a)-(3log_{2}b+2log_{2}c) \\\\ \boxed{\boxed{3+log_{2}a-3log_{2}b-2log_{2}c}}$