Question

Dado ln 2=0,69 e ln 3=1,09 determine o valor da raiz de cada equação abaixo:

Answer 1

Resposta:

a) x=0,69

b) x=0,89

c) x=1,78

d) x=24,7

Explicação passo-a-passo:

a)

$\displaystylee^x=2\\lne^x=ln2\\x.lne=0,69\\x=0,69$

b)

$\displaystyle\\e^{2x}=6\\lne^{2x}=ln6\\2x=ln(3.2)=ln3+ln2=1,09+0,69=1,78\\\\x=\frac{1,78}{2} =0,89$

c)

$e^{2x}=36\\lne^{2x}=ln36\\2x=ln(6)^2=2.ln(3.2)\\x=ln3+ln2=1,09+0,69=1,78$

d)

$e^{-0,1x}=\frac{1}{12}\\\\e^{-0,1x}=12^{-1}\\lne^{-0,1x}=ln12^{-1}\\-0,1x=-ln12\\x=10.ln12=10.ln(4.3)=10.(ln4+ln3)=10.(ln2^2+1,09)=10.(2.ln2+1,09)=10.(2.0,69+1,09)=10.(1,38+1,09)=10.(2,47)=24,7$

obs:

lne=1

$Propriedades:\\\\(a^{m})^{n}=a^{m.n}\\\\\sqrt[n]{x^m} =x^{\frac{m}{n} }\\\\a^{m}a^{n}=a^{m+n}\\\\\frac{a^{m}}{a^{n}}=a^{m-n} \\\\a^{0}=1\\\\a^{1}=a\\\\ln(ab)=ln(a)+ln(b)\\\\ln(a)^b=b.ln(a)$