determine o tempo que um capital aplicado a juros compostos a taxa de 20% ao ano leva para triplicar o seu valor (use log2=0,3 e long3=0,48)
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AE manoo,
temos..
e relembrando as propriedades de logaritmos, da potência, do quociente, do produto e a decorrente da definição (D2)..
podemos então aplicar a fórmula geral dos juros compostos para o tempo (t períodos)..
![M=C(1+i)^t\\\\
3\cdot C=C\cdot(1+0,2)^t\\
(1,2)^t=3\\
\log(1,2)^t=\log(3)\\\\
\log\left( \dfrac{12}{10}\right)^t=\log(3)\\\\
\log_{10}\left( \dfrac{2^2\cdot3}{10}\right)^t=\log_{10}(3)\\\\
\{[\log_{10}(2)^2+\log_{10}(3)]-\log_{10}(10)\}^t=\log_{10}(3)\\\\
t\cdot\{2\cdot[\log_{10}(2)+\log_{10}(3)]-\log_{10}(10)\}=\log_{10}(3)](https://tex.z-dn.net/?f=M%3DC%281%2Bi%29%5Et%5C%5C%5C%5C%0A3%5Ccdot+C%3DC%5Ccdot%281%2B0%2C2%29%5Et%5C%5C%0A%281%2C2%29%5Et%3D3%5C%5C%0A%5Clog%281%2C2%29%5Et%3D%5Clog%283%29%5C%5C%5C%5C%0A%5Clog%5Cleft%28+%5Cdfrac%7B12%7D%7B10%7D%5Cright%29%5Et%3D%5Clog%283%29%5C%5C%5C%5C%0A%5Clog_%7B10%7D%5Cleft%28+%5Cdfrac%7B2%5E2%5Ccdot3%7D%7B10%7D%5Cright%29%5Et%3D%5Clog_%7B10%7D%283%29%5C%5C%5C%5C%0A%5C%7B%5B%5Clog_%7B10%7D%282%29%5E2%2B%5Clog_%7B10%7D%283%29%5D-%5Clog_%7B10%7D%2810%29%5C%7D%5Et%3D%5Clog_%7B10%7D%283%29%5C%5C%5C%5C%0At%5Ccdot%5C%7B2%5Ccdot%5B%5Clog_%7B10%7D%282%29%2B%5Clog_%7B10%7D%283%29%5D-%5Clog_%7B10%7D%2810%29%5C%7D%3D%5Clog_%7B10%7D%283%29++ "M=C(1+i)^t\\\\
3\cdot C=C\cdot(1+0,2)^t\\
(1,2)^t=3\\
\log(1,2)^t=\log(3)\\\\
\log\left( \dfrac{12}{10}\right)^t=\log(3)\\\\
\log_{10}\left( \dfrac{2^2\cdot3}{10}\right)^t=\log_{10}(3)\\\\
\{[\log_{10}(2)^2+\log_{10}(3)]-\log_{10}(10)\}^t=\log_{10}(3)\\\\
t\cdot\{2\cdot[\log_{10}(2)+\log_{10}(3)]-\log_{10}(10)\}=\log_{10}(3)")
![\log(2)=0,3;~~\log(3)=0,48~~e~~\log_{10}(10)=1\\\\t\cdot\{2\cdot[0,3+0,48]-1\}=0,48\\
t\cdot\{2\cdot[0,78]-1\}=0,48\\
t\cdot\{1,56-1\}=0,48\\
t\cdot0,56=0,48\\\\
t= \dfrac{0,48}{0,56}\\\\
t\approx0,8~~(transformando~em~meses)..(\times12)\\\\
t\approx9,6~meses~~(transformando~0,6,~em~dias)..\\\\
\Large\boxed{t\approx9~meses~e~18~dias}](https://tex.z-dn.net/?f=%5Clog%282%29%3D0%2C3%3B%7E%7E%5Clog%283%29%3D0%2C48%7E%7Ee%7E%7E%5Clog_%7B10%7D%2810%29%3D1%5C%5C%5C%5Ct%5Ccdot%5C%7B2%5Ccdot%5B0%2C3%2B0%2C48%5D-1%5C%7D%3D0%2C48%5C%5C%0At%5Ccdot%5C%7B2%5Ccdot%5B0%2C78%5D-1%5C%7D%3D0%2C48%5C%5C%0At%5Ccdot%5C%7B1%2C56-1%5C%7D%3D0%2C48%5C%5C%0At%5Ccdot0%2C56%3D0%2C48%5C%5C%5C%5C%0At%3D+%5Cdfrac%7B0%2C48%7D%7B0%2C56%7D%5C%5C%5C%5C%0At%5Capprox0%2C8%7E%7E%28transformando%7Eem%7Emeses%29..%28%5Ctimes12%29%5C%5C%5C%5C%0At%5Capprox9%2C6%7Emeses%7E%7E%28transformando%7E0%2C6%2C%7Eem%7Edias%29..%5C%5C%5C%5C%0A%5CLarge%5Cboxed%7Bt%5Capprox9%7Emeses%7Ee%7E18%7Edias%7D+ "\log(2)=0,3;~~\log(3)=0,48~~e~~\log_{10}(10)=1\\\\t\cdot\{2\cdot[0,3+0,48]-1\}=0,48\\
t\cdot\{2\cdot[0,78]-1\}=0,48\\
t\cdot\{1,56-1\}=0,48\\
t\cdot0,56=0,48\\\\
t= \dfrac{0,48}{0,56}\\\\
t\approx0,8~~(transformando~em~meses)..(\times12)\\\\
t\approx9,6~meses~~(transformando~0,6,~em~dias)..\\\\
\Large\boxed{t\approx9~meses~e~18~dias}")
Tenha ótimos estudos ;D
temos..
e relembrando as propriedades de logaritmos, da potência, do quociente, do produto e a decorrente da definição (D2)..
podemos então aplicar a fórmula geral dos juros compostos para o tempo (t períodos)..
Tenha ótimos estudos ;D
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