me ajudem em uma questao de matemática
Sejam a e b as raízes da equação x²- 100x + 1000 = 0 . Determine o valor de :
Log a + Log b - Log b - Log (a+b)
não consegui fazer esta questão podem me ajudar
didisouzarj:
por favor
correção
Faça a correção!
log a + log b - log (a+b)
desculpa
Veja se consegue entender!
Respostas
respondido por:
0
Boa noite Didi!
Solução!
![x^{2} -100x+1000=0\\\\\\\
x= \dfrac{100\pm \sqrt{(-100)^{2}-4.1.1000 } }{2} \\\\\\\\\
x= \dfrac{100\pm \sqrt{10000-4000 } }{2} \\\\\\\\\\
x= \dfrac{100\pm \sqrt{6000} }{2} \\\\\\\\\\
x= \dfrac{100\pm 20\sqrt{15} }{2} \\\\\\\\\\
x= \dfrac{2(50\pm 10\sqrt{15} )}{2} \\\\\\\\\\
x= (50\pm 10\sqrt{15} )\\\\\\\\\\
a=50+ \sqrt{15}\\\\\
b=50- \sqrt{15}](https://tex.z-dn.net/?f=+x%5E%7B2%7D+-100x%2B1000%3D0%5C%5C%5C%5C%5C%5C%5C%0Ax%3D+%5Cdfrac%7B100%5Cpm+%5Csqrt%7B%28-100%29%5E%7B2%7D-4.1.1000+%7D+%7D%7B2%7D+%5C%5C%5C%5C%5C%5C%5C%5C%5C%0Ax%3D+%5Cdfrac%7B100%5Cpm+%5Csqrt%7B10000-4000+%7D+%7D%7B2%7D+%5C%5C%5C%5C%5C%5C%5C%5C%5C%5C%0Ax%3D+%5Cdfrac%7B100%5Cpm+%5Csqrt%7B6000%7D+%7D%7B2%7D+%5C%5C%5C%5C%5C%5C%5C%5C%5C%5C%0Ax%3D+%5Cdfrac%7B100%5Cpm+20%5Csqrt%7B15%7D+%7D%7B2%7D+%5C%5C%5C%5C%5C%5C%5C%5C%5C%5C%0Ax%3D+%5Cdfrac%7B2%2850%5Cpm+10%5Csqrt%7B15%7D+%29%7D%7B2%7D+%5C%5C%5C%5C%5C%5C%5C%5C%5C%5C%0Ax%3D+%2850%5Cpm+10%5Csqrt%7B15%7D+%29%5C%5C%5C%5C%5C%5C%5C%5C%5C%5C%0A%0Aa%3D50%2B+%5Csqrt%7B15%7D%5C%5C%5C%5C%5C%0Ab%3D50-+%5Csqrt%7B15%7D++%0A%0A "x^{2} -100x+1000=0\\\\\\\
x= \dfrac{100\pm \sqrt{(-100)^{2}-4.1.1000 } }{2} \\\\\\\\\
x= \dfrac{100\pm \sqrt{10000-4000 } }{2} \\\\\\\\\\
x= \dfrac{100\pm \sqrt{6000} }{2} \\\\\\\\\\
x= \dfrac{100\pm 20\sqrt{15} }{2} \\\\\\\\\\
x= \dfrac{2(50\pm 10\sqrt{15} )}{2} \\\\\\\\\\
x= (50\pm 10\sqrt{15} )\\\\\\\\\\
a=50+ \sqrt{15}\\\\\
b=50- \sqrt{15}")
![loga+logb-log(a+b)\\\\\\\
log(50+10 \sqrt{15}) +log(50-10 \sqrt{15})-log(50+10 \sqrt{15} +50-10 \sqrt{15})\\\\\\\
log[(50+10 \sqrt{15}) \times(50-10 \sqrt{15})]-log(2.(50)\\\\\\\
log(2500-500 \sqrt{15}+500 \sqrt{15}-(10 \sqrt{15})^{2} -log(100)\\\\\\\\\
log(2500-(10 \sqrt{15})^{2})-log(100)\\\\\\\
log(2500-100.15)-log(100)\\\\\\
log(2500-1500)-log(100)\\\\\
log1000-log100](https://tex.z-dn.net/?f=loga%2Blogb-log%28a%2Bb%29%5C%5C%5C%5C%5C%5C%5C%0Alog%2850%2B10+%5Csqrt%7B15%7D%29+%2Blog%2850-10+%5Csqrt%7B15%7D%29-log%2850%2B10+%5Csqrt%7B15%7D+%2B50-10+%5Csqrt%7B15%7D%29%5C%5C%5C%5C%5C%5C%5C%0A%0Alog%5B%2850%2B10+%5Csqrt%7B15%7D%29+%5Ctimes%2850-10+%5Csqrt%7B15%7D%29%5D-log%282.%2850%29%5C%5C%5C%5C%5C%5C%5C%0Alog%282500-500+%5Csqrt%7B15%7D%2B500+%5Csqrt%7B15%7D-%2810+%5Csqrt%7B15%7D%29%5E%7B2%7D+-log%28100%29%5C%5C%5C%5C%5C%5C%5C%5C%5C%0Alog%282500-%2810+%5Csqrt%7B15%7D%29%5E%7B2%7D%29-log%28100%29%5C%5C%5C%5C%5C%5C%5C%0Alog%282500-100.15%29-log%28100%29%5C%5C%5C%5C%5C%5C%0Alog%282500-1500%29-log%28100%29%5C%5C%5C%5C%5C%0Alog1000-log100+++++++%0A++ "loga+logb-log(a+b)\\\\\\\
log(50+10 \sqrt{15}) +log(50-10 \sqrt{15})-log(50+10 \sqrt{15} +50-10 \sqrt{15})\\\\\\\
log[(50+10 \sqrt{15}) \times(50-10 \sqrt{15})]-log(2.(50)\\\\\\\
log(2500-500 \sqrt{15}+500 \sqrt{15}-(10 \sqrt{15})^{2} -log(100)\\\\\\\\\
log(2500-(10 \sqrt{15})^{2})-log(100)\\\\\\\
log(2500-100.15)-log(100)\\\\\\
log(2500-1500)-log(100)\\\\\
log1000-log100")
![log1000\\\\\\
log1000=x\\\\\\\
10^{x}=1000\\\\\
10^{x}=10^{3}\\\\\\
\boxed{x=3}\\\\\\\\
log100\\\\\\
log100=x\\\\\\
10^{x} =100\\\\\\
10^{x}=10^{2}\\\\\\
\boxed{x=2}](https://tex.z-dn.net/?f=log1000%5C%5C%5C%5C%5C%5C%0Alog1000%3Dx%5C%5C%5C%5C%5C%5C%5C%0A10%5E%7Bx%7D%3D1000%5C%5C%5C%5C%5C%0A10%5E%7Bx%7D%3D10%5E%7B3%7D%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7Bx%3D3%7D%5C%5C%5C%5C%5C%5C%5C%5C%0Alog100%5C%5C%5C%5C%5C%5C%0Alog100%3Dx%5C%5C%5C%5C%5C%5C%0A10%5E%7Bx%7D+%3D100%5C%5C%5C%5C%5C%5C%0A10%5E%7Bx%7D%3D10%5E%7B2%7D%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7Bx%3D2%7D+++++ "log1000\\\\\\
log1000=x\\\\\\\
10^{x}=1000\\\\\
10^{x}=10^{3}\\\\\\
\boxed{x=3}\\\\\\\\
log100\\\\\\
log100=x\\\\\\
10^{x} =100\\\\\\
10^{x}=10^{2}\\\\\\
\boxed{x=2}")
![log1000-log100\\\\\
~~~~3~~~~-~~~~2=1](https://tex.z-dn.net/?f=log1000-log100%5C%5C%5C%5C%5C%0A%7E%7E%7E%7E3%7E%7E%7E%7E-%7E%7E%7E%7E2%3D1 "log1000-log100\\\\\
~~~~3~~~~-~~~~2=1")
Boa noite!
Bons estudos!
Solução!
Boa noite!
Bons estudos!
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