Question

sabendo que log 2= a e log 3= b, obtenha, em função de a e b, o valor de:
a)log₆ 5
b)log₁₀₀ 36
c)log₄ 18

Answer 1

$log\ _6\ 5 = \frac{log\ 5}{log\ 6}\\\\\ log\ 5 = log\ \frac{10}{2} = log\ 10-log\ 2 = 1-a\\\\ log\ 6 = log\ 2*3= log\ 2+log\ 3 = a+b\\\\\ log\ _6\ 5 = \frac{log\ 5}{log\ 6} = \frac{1-a}{a+b}\\\\ \boxed{log\ _6\ 5= \frac{1-a}{a+b}}$

$log\ _1_0_0\ 36 = \frac{log\ 36}{log\ 100}\\\\\ log\ 36 = log\ 2^2*3^2 =2log\ 2+2\ log\ 3 = 2a+2b\\\\ log\ 100 = 2\\\\ log\ _1_0_0\ 36 = \frac{log\ 36}{log\ 100} = \frac{2a+2b}{2}\\\\ \boxed{log\ _1_0_0\ 36 = a+b}}$

$log\ _4\ 18 =\frac{log\ 18}{log\ 4}\\\\\ log\ 18 = log\ 2*3^2 = log\ 2+2log\ 3 = a+2b\\\\ log\ 4 = log\ 2^2 = 2\ log\ 2 = 2a\\\\\ log\ _4\ 18 =\frac{log\ 18}{log\ 4} = \frac{a+2b}{2a}\\\\ \boxed{log\ _4\ 18 = \frac{a+2b}{2a}}$