Determine o conjunto de valores reais de x que satisfazem cada uma das equações:
a) 2^x+1 + 2^x−1 = 20
b) log base2(x + 1) + log base2(x − 1) = 3
Respostas
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Boa tarde!
Solução!
![2^{x+1} +2^{x-1}=20\\\\\\
2^{x}.2+2^{x}.2^{-1}=20\\\\\\
2^{x}(2+2^{-1})=20\\\\\\\
2^{x}(2+ \frac{1}{2})=20 \\\\\\\
2^{x}( \frac{4+1}{2})=20\\\\\\
2^{x}( \frac{5}{2})=20\\\\\
2^{x}= \dfrac{20}{ \dfrac{5}{2} } \\\\\\
2^{x}=20. \dfrac{2}{5}\\\\\
2^{x} =4.2\\\\\\
2^{x} =8\\\\\
2^{x}=2^{3}\\\\\\
\boxed{x=3}](https://tex.z-dn.net/?f=2%5E%7Bx%2B1%7D+%2B2%5E%7Bx-1%7D%3D20%5C%5C%5C%5C%5C%5C%0A2%5E%7Bx%7D.2%2B2%5E%7Bx%7D.2%5E%7B-1%7D%3D20%5C%5C%5C%5C%5C%5C%0A2%5E%7Bx%7D%282%2B2%5E%7B-1%7D%29%3D20%5C%5C%5C%5C%5C%5C%5C%0A2%5E%7Bx%7D%282%2B+%5Cfrac%7B1%7D%7B2%7D%29%3D20+%5C%5C%5C%5C%5C%5C%5C%0A2%5E%7Bx%7D%28+%5Cfrac%7B4%2B1%7D%7B2%7D%29%3D20%5C%5C%5C%5C%5C%5C%0A2%5E%7Bx%7D%28+%5Cfrac%7B5%7D%7B2%7D%29%3D20%5C%5C%5C%5C%5C%0A2%5E%7Bx%7D%3D+%5Cdfrac%7B20%7D%7B+%5Cdfrac%7B5%7D%7B2%7D+%7D+%5C%5C%5C%5C%5C%5C%0A2%5E%7Bx%7D%3D20.+%5Cdfrac%7B2%7D%7B5%7D%5C%5C%5C%5C%5C%0A2%5E%7Bx%7D+%3D4.2%5C%5C%5C%5C%5C%5C%0A2%5E%7Bx%7D+%3D8%5C%5C%5C%5C%5C%0A2%5E%7Bx%7D%3D2%5E%7B3%7D%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7Bx%3D3%7D "2^{x+1} +2^{x-1}=20\\\\\\
2^{x}.2+2^{x}.2^{-1}=20\\\\\\
2^{x}(2+2^{-1})=20\\\\\\\
2^{x}(2+ \frac{1}{2})=20 \\\\\\\
2^{x}( \frac{4+1}{2})=20\\\\\\
2^{x}( \frac{5}{2})=20\\\\\
2^{x}= \dfrac{20}{ \dfrac{5}{2} } \\\\\\
2^{x}=20. \dfrac{2}{5}\\\\\
2^{x} =4.2\\\\\\
2^{x} =8\\\\\
2^{x}=2^{3}\\\\\\
\boxed{x=3}")
![log_{2}(x+1)+log_{2}(x-1)=3\\\\\\
log_{2}[(x+1)+(x-1)]=3\\\\\\
(x+1)\times(x-1)=2^{3} \\\\\\
x^{2} +x-x-1=8\\\\\\\
x^{2} -1=8\\\\\\\
x^{2} =8+1\\\\\
x^{2} =9\\\\\
x= \sqrt{9} \\\\\\
x=\pm3\\\\\\\\\
\boxed{Obs: -3~~n\~ao~~serve:~~n\~ao~~existe~~log~~de~~um~~numero~~negativo }\\\\\\\
\boxed{Resposta:x=3}](https://tex.z-dn.net/?f=log_%7B2%7D%28x%2B1%29%2Blog_%7B2%7D%28x-1%29%3D3%5C%5C%5C%5C%5C%5C%0Alog_%7B2%7D%5B%28x%2B1%29%2B%28x-1%29%5D%3D3%5C%5C%5C%5C%5C%5C%0A%28x%2B1%29%5Ctimes%28x-1%29%3D2%5E%7B3%7D+%5C%5C%5C%5C%5C%5C%0A+x%5E%7B2%7D+%2Bx-x-1%3D8%5C%5C%5C%5C%5C%5C%5C%0A+x%5E%7B2%7D+-1%3D8%5C%5C%5C%5C%5C%5C%5C%0A+x%5E%7B2%7D+%3D8%2B1%5C%5C%5C%5C%5C%0A+x%5E%7B2%7D+%3D9%5C%5C%5C%5C%5C%0Ax%3D+%5Csqrt%7B9%7D+%5C%5C%5C%5C%5C%5C%0Ax%3D%5Cpm3%5C%5C%5C%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7BObs%3A+-3%7E%7En%5C%7Eao%7E%7Eserve%3A%7E%7En%5C%7Eao%7E%7Eexiste%7E%7Elog%7E%7Ede%7E%7Eum%7E%7Enumero%7E%7Enegativo+%7D%5C%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7BResposta%3Ax%3D3%7D+ "log_{2}(x+1)+log_{2}(x-1)=3\\\\\\
log_{2}[(x+1)+(x-1)]=3\\\\\\
(x+1)\times(x-1)=2^{3} \\\\\\
x^{2} +x-x-1=8\\\\\\\
x^{2} -1=8\\\\\\\
x^{2} =8+1\\\\\
x^{2} =9\\\\\
x= \sqrt{9} \\\\\\
x=\pm3\\\\\\\\\
\boxed{Obs: -3~~n\~ao~~serve:~~n\~ao~~existe~~log~~de~~um~~numero~~negativo }\\\\\\\
\boxed{Resposta:x=3}")
Boa tarde!
Bons estudos!
Solução!
Boa tarde!
Bons estudos!
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