Question

Considerando log 2=0,3, log3 = 0,5, log5=0,7

a) Log 15
b) Log45
c) Log30
d) Log 1,2
e)log√0,3

Answer 1

Explicação passo-a-passo:

Lembre-se que:

• $\sf log_{b}~(a\cdot c)=log_{b}~a+log_{b}~c$

• $\sf log_{b}~\Big(\dfrac{a}{c}\Big)=log_{b}~a-log_{b}~c$

• $\sf log_{b}~a^m=m\cdot log_{b}~a$

• $\sf \sqrt[c]{a^b}=a^{\frac{b}{c}}$

a)

$\sf log~15=log~(3\cdot5)$

$\sf log~15=log~3+log~5$

$\sf log~15=0,5+0,7$

$\sf \red{log~15=1,2}$

b)

$\sf log~45=log~(3^2\cdot5)$

$\sf log~45=log~3^2+log~5$

$\sf log~45=2\cdot log~3+log~5$

$\sf log~45=2\cdot0,5+0,7$

$\sf log~45=1+0,7$

$\sf \red{log~45=1,7}$

c)

$\sf log~30=log~(2\cdot3\cdot5)$

$\sf log~30=log~2+log~3+log~5$

$\sf log~30=0,3+0,5+0,7$

$\sf \red{log~30=1,5}$

d)

$\sf log~1,2=log~\Big(\dfrac{12}{10}\Big)$

$\sf log~1,2=log~\Big(\dfrac{6}{5}\Big)$

$\sf log~1,2=log~6-log~5$

$\sf log~1,2=log~(2\cdot3)-log~5$

$\sf log~1,2=log~2+log~3-log~5$

$\sf log~1,2=0,3+0,5-0,7$

$\sf \red{log~1,2=0,1}$

e)

$\sf log~\sqrt{0,3}=log~\sqrt{\dfrac{3}{10}}$

$\sf log~\sqrt{0,3}=log~\Big(\dfrac{3}{10}\Big)^{\frac{1}{2}}$

$\sf log~\sqrt{0,3}=\dfrac{1}{2}\cdot log~\Big(\dfrac{3}{10}\Big)$

$\sf log~\sqrt{0,3}=\dfrac{1}{2}\cdot(log~3-log~10)$

$\sf log~\sqrt{0,3}=\dfrac{1}{2}\cdot(0,5-1)$

$\sf log~\sqrt{0,3}=\dfrac{1}{2}\cdot(-0,5)$

$\sf log~\sqrt{0,3}=\dfrac{1}{2}\cdot\Big(-\dfrac{1}{2}\Big)$

$\sf \red{log~\sqrt{0,3}=-\dfrac{1}{4}}$