Question

Aplicando as Propriedades do Logaritmo Desenvolva:

a)
$log_{2} \bigg( \dfrac{2ab}{c} \bigg)$

b)
$log \bigg( \dfrac{ {a}^{3}. {b}^{2} }{ {c}^{4} } \bigg)$
​

Answer 1

a) Usando:

loga (x/y) = loga (x) - loga (y)

$log_{2}( \frac{2ab}{c} )$

$log_{2}(2ab) - log_{2}(c)$

$log_{2}(2) + log_{2}(a) + log_{2}(b) - log_{2}(c)$

$1 + log_{2}(a) + log_{2}(b) - log_{2}(c)$

b)

$log( \frac{x}{y} ) = log(x) - log(y)$

$log( \frac{ {a}^{3} \times {b}^{2} }{ {c}^{4} } )$

$(3 log(a) + 2 log(b) ) - 4 log(c)$

Answer 2

$\sf \bullet A) \to \boxed{\boxed{\sf =1+log _2\left(a\right)+log _2\left(b\right)-log _2\left(c\right)}}$

$\tt EXPLICA \c{C}AO\\\\\\\sf log _2\left(\dfrac{2ab}{c}\right)\\\\\\\sf \displaystyle \bullet {Aplicar\:as\:propriedades\:dos\:logaritmos}:\quad log _c\left(\frac{a}{b}\right)=log _c\left(a\right)-log _c\left(b\right)\\\\\\\sf log _2\left(\frac{2ab}{c}\right)=log _2\left(2ab\right)-log _2\left(c\right)\\\\\\\sf =log _2\left(2ab\right)-log _2\left(c\right)\\\\\\\sf \bullet log _c\left(ab\right)=log _c\left(a\right)+log _c\left(b\right)$

$\sf log _2\left(2ab\right)=log _2\left(2\right)+log _2\left(a\right)+log _2\left(b\right)\\\\\\\sf =log _2\left(2\right)+log _2\left(a\right)+log _2\left(b\right)-log _2\left(c\right)\\\\\\\sf log _a\left(a\right)=1\\\\\\\boxed{\boxed{\sf =1+log _2\left(a\right)+log _2\left(b\right)-log _2\left(c\right)}}$

$\sf \bullet B) \to \boxed{\boxed{\sf =3log _{10}\left(a\right)+2log _{10}\left(b\right)-4log _{10}\left(c\right)}}$

$\tt Vamos \ novamente\\\\\\\sf \displaystyle log\left(\frac{a^3\cdot \:b^2}{c^4}\right)\\\\\\\sf log _{10}\left(\frac{a^3b^2}{c^4}\right)\\\\\\\large{\star} \ log _c\left(\frac{a}{b}\right)=log _c\left(a\right)-log _c\left(b\right)\\\\\\\sf log _{10}\left(\frac{a^3b^2}{c^4}\right)=log _{10}\left(a^3b^2\right)-log _{10}\left(c^4\right)\\\\\\\sf =log _{10}\left(a^3b^2\right)-log _{10}\left(c^4\right)\\\\\\\sf log _{10}\left(c^4\right)=4log _{10}\left(c\right)$

$\sf \displaystyle =log _{10}\left(a^3b^2\right)-4log _{10}\left(c\right)\\\\\\\sf \star Regra \ dos \ logaritmos \iff log _c\left(ab\right)=log _c\left(a\right)+log _c\left(b\right)\\\\\\\sf log _{10}\left(a^3b^2\right)=log _{10}\left(a^3\right)+log _{10}\left(b^2\right)\\\\\\\sf =log _{10}\left(a^3\right)+log _{10}\left(b^2\right)-4log _{10}\left(c\right)\\\\\\\sf \star \ log _a\left(x^b\right)=b\cdot log _a\left(x\right)\\\\\\\sf log _{10}\left(a^3\right)=3log _{10}\left(a\right)$

$\sf log _{10}\left(a^3\right)=3log _{10}\left(a\right)\\\\\\\sf =3log _{10}\left(a\right)+log _{10}\left(b^2\right)-4log _{10}\left(c\right)\\\\\\\sf \bullet log _a\left(x^b\right)=b\cdot log _a\left(x\right)\\\\\\\sf log _{10}\left(b^2\right)=2log _{10}\left(b\right)\\\\\\\sf \to \boxed{\boxed{\sf =3log _{10}\left(a\right)+2log _{10}\left(b\right)-4log _{10}\left(c\right)}}$