Exercícios sobre propriedades dos logaritmos

Sendo a , b e c números reais positivos, desenvolver as expressões abaixo.

$\;log_{{}_{\Large \,2\,}}\left(\dfrac{\,2ab\,}{c}\right)\,$

$\;log_{{}_{\Large \,3\,}}\left(\dfrac{\,a^{\large 3}b^{\large 2}\,}{c^{\large 4}}\right)\,$

$\;log\,\left(\dfrac{\,a^{\large 3}\,}{\,b^{{}^{\Large 2}}\,\centerdot\,\sqrt{\,c\,}\,}\right)\,$

resposta: a) $1\,+\,log_2\,a\,+\,log_2\,b\,-\,log_2\,c$ b) $3\,log_3\,a\,+\,2\,log_3\,b\,-\,4\,log_3\,c$ c) $\,3\,log\,a\,-\,2\,log\,b\,-\,\dfrac{1}{2}\,log\,c$
×

Desenvolver as expressões abaixo aplicando as propriedades dos logaritmos.

$\;log_{{}_{\Large \,5\,}}(\dfrac{\,5a\,}{bc})\,$

$\;log_{{}_{\Large \,3\,}}(\dfrac{\,ab^2\,}{c})\,$

$\;log_{{}_{\Large \,2\,}}\left(\dfrac{\,a^2\,\sqrt{\,b\,}\,}{\sqrt[\large \,3\,]{\,c\,}} \right) $

$\;log_{{}_{\Large \,3\,}}\left(\dfrac{\,a\,\centerdot\,b^3\,}{c\,\centerdot\,\sqrt[\large \,3\,]{\,a^2\,}}\right)\,$

$\;log\sqrt{\dfrac{\,ab^3\,}{c^2}}\,$

$\;log_{{}_{\Large \,3\,}}\sqrt[\Large 3\,]{\dfrac{\,a\,}{\,b^2\,\centerdot\,\sqrt{\,c\,}}}\,$

$\;log_{{}_{\Large \,2\,}}\sqrt{\dfrac{\,4a\,\sqrt{\,ab\,}}{\,b\;\sqrt[\Large 3\,]{\,a^2b\,}}}\,$

$\;log\,\left(\sqrt[\LARGE 3\,]{\dfrac{\,a^{\large 4}\,\sqrt{\,ab\,}}{\,b^2\;\sqrt[\Large 3\,]{\,bc\,}}}\right)^{\Large 2}\,$

resposta: a) $\,1\,+\,log_5a\,-\,log_5b\,-log_5c\,$ b) $\,log_3a\,+\,2\,log_3b\,-\,log_3c\,$ c) $2\,log_2a\,+\,\frac{1}{2}log_2b\,-\,\frac{1}{3}log_2c$ d) $\,\frac{1}{3}\,log_3a\,+\,3\,log_3b\,-\,log_3c$ e) $\frac{1}{2}\,log\,a\,+\,\frac{3}{2}log\,b\,-\,log\,c$ f) $\frac{1}{3}\,log\,a\,-\,\frac{2}{3}log\,b\,-\,\frac{1}{6}log\,c$ g) $\,2\,+\,\frac{5}{12}log_2a\,-\,\frac{5}{12}log_2b\,$h) $\,3\,log\,a\,-\,\frac{11}{9}log\,b\,-\,\frac{2}{9}log\,c\,$
×

Calcular:
a) $\,antilog_{\,2\,}(log_2\,3)\;$ b)$\,antilog_{\,3\,}(log_3\,5)\,$

resposta: a) 3 b) 5
×

Desenvolver aplicando as propriedades dos logaritmos. Obs. a > b > c > 0 .

$\;log_{{}_{\Large \,2\,}}\dfrac{2a}{\;a^2\,-\,b^2\;}\;$

$\;log_{{}_{\Large \,2\,}}\dfrac{a^2\,\sqrt{\,bc\,}}{\;\sqrt[\LARGE 5]{\,(a\,+\,b)^3}\;}\;$

$\;log\left(c\,\centerdot\,\sqrt[\LARGE 3]{\dfrac{\;a(a\,+\,b)^2}{\sqrt{\;b\;}}} \right)\;$

$\;log\left(\dfrac{\;\sqrt[\Large 5]{a(a\,-\,b)^2}\;}{\sqrt{a^2\,+\,b^2}} \right)\;$

resposta: a) b) c) d)
×

Sendo a, b e c reais positivos, escreva as expressões cujos desenvolvimentos logaritmicos são dados.

$\;log_{{}_{\Large \,2\,}}a\,+\,log_{{}_{\Large \,2\,}}b\,-\,log_{{}_{\Large \,2\,}}c\;$

$\;2\,log\,a\;-\;log\,b\;-\;3\,log\,c\;$

$\;2\,-\,log_{{}_{\Large \,3\,}}a\,+\,3\,log_{{}_{\Large \,3\,}}b\,-\,2\,log_{{}_{\Large \,3\,}}c\;$

$\;\dfrac{\;1\;}{2}\,log\;a\,-\;2\,log\,b\;-\;\dfrac{\;1\;}{3}\,log\,c\;$

$\;\dfrac{\;1\;}{3}\,log\;a\,-\;\dfrac{\;1\;}{2}\,log\,c\;-\;\dfrac{\;3\;}{2}\,log\,b\;$

$\;2\;+\;\dfrac{\;\,1\,\;}{3}\,log_{{}_{\Large \,2\,}}a\,+\,\dfrac{\;\,1\,\;}{6}\,log_{{}_{\Large \,2\,}}b\,-\,log_{{}_{\Large \,2\,}}c\;$

$\;\dfrac{\;1\;}{4}(log\,a\;-\;3\,log\,b\;-\;2\,log\,c)\;$

resposta: a) b) c) d)
×

Se $\;log\,2\;=\;a\phantom{X}$ e $\phantom{X}log\,3\;=\;b\;$, colocar em função de $\,a\,$ e $\,b\,$ os seguintes logaritmos decimais:

$\,log\,6\,$

$\,log\,4\,$

$\,log\,12\,$

$\,log\,\sqrt{\,2\,}\,$

$\,log\,0,5\,$

$\,log\,20\,$

$\,log\,5\,$

$\,log\,15\,$

resposta: a) b) c) d)
×