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log(2cosx)/log3=4 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
log(2*cos(x))    
------------- = 4
    log(3)
log⁡(2cos⁡(x))log⁡(3)=4\frac{\log{\left(2 \cos{\left(x \right)} \right)}}{\log{\left(3 \right)}} = 4
Simplificar
Solución detallada
Tenemos la ecuación
log⁡(2cos⁡(x))log⁡(3)=4\frac{\log{\left(2 \cos{\left(x \right)} \right)}}{\log{\left(3 \right)}} = 4
cambiamos
log⁡(2cos⁡(x)81)log⁡(3)=0\frac{\log{\left(\frac{2 \cos{\left(x \right)}}{81} \right)}}{\log{\left(3 \right)}} = 0
log⁡(2cos⁡(x))log⁡(3)−4=0\frac{\log{\left(2 \cos{\left(x \right)} \right)}}{\log{\left(3 \right)}} - 4 = 0
Sustituimos
w=cos⁡(x)w = \cos{\left(x \right)}
Tenemos la ecuación
log⁡(2w)log⁡(3)−4=0\frac{\log{\left(2 w \right)}}{\log{\left(3 \right)}} - 4 = 0
log⁡(2w)log⁡(3)=4\frac{\log{\left(2 w \right)}}{\log{\left(3 \right)}} = 4
Devidimos ambás partes de la ecuación por el multiplicador de log =1/log(3)
log⁡(2w)=4log⁡(3)\log{\left(2 w \right)} = 4 \log{\left(3 \right)}
Es la ecuación de la forma:
log(v)=p

Por definición log
v=e^p

entonces
2w=e41log⁡(3)2 w = e^{\frac{4}{\frac{1}{\log{\left(3 \right)}}}}
simplificamos
2w=812 w = 81
w=812w = \frac{81}{2}
hacemos cambio inverso
cos⁡(x)=w\cos{\left(x \right)} = w
Tenemos la ecuación
cos⁡(x)=w\cos{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
O
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
, donde n es cualquier número entero
sustituimos w:
Gráfica
0-80-60-40-2020406080-100100-1010
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
x1 = 2*pi - I*im(acos(81/2))
x1=2π−iim⁡(acos⁡(812))x_{1} = 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)} 
x2 = I*im(acos(81/2)) + re(acos(81/2))
x2=re⁡(acos⁡(812))+iim⁡(acos⁡(812))x_{2} = \operatorname{re}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)} 
x2 = re(acos(81/2)) + i*im(acos(81/2))
Suma y producto de raíces [src] 
suma
2*pi - I*im(acos(81/2)) + I*im(acos(81/2)) + re(acos(81/2))
(2π−iim⁡(acos⁡(812)))+(re⁡(acos⁡(812))+iim⁡(acos⁡(812)))\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)}\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)}\right) 
=
2*pi + re(acos(81/2))
re⁡(acos⁡(812))+2π\operatorname{re}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)} + 2 \pi 
producto
(2*pi - I*im(acos(81/2)))*(I*im(acos(81/2)) + re(acos(81/2)))
(2π−iim⁡(acos⁡(812)))(re⁡(acos⁡(812))+iim⁡(acos⁡(812)))\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)}\right) 
=
(2*pi - I*im(acos(81/2)))*(I*im(acos(81/2)) + re(acos(81/2)))
(2π−iim⁡(acos⁡(812)))(re⁡(acos⁡(812))+iim⁡(acos⁡(812)))\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{81}{2} \right)}\right)}\right) 
(2*pi - i*im(acos(81/2)))*(i*im(acos(81/2)) + re(acos(81/2)))
Respuesta numérica [src] 
x1 = -56.5486677646163 + 4.3942967040245*i
x2 = -150.79644737231 + 4.3942967040245*i
x3 = 31.4159265358979 - 4.3942967040245*i
x4 = -12.5663706143592 - 4.3942967040245*i
x5 = 87.9645943005142 + 4.3942967040245*i
x6 = 56.5486677646163 + 4.3942967040245*i
x7 = 75.398223686155 + 4.3942967040245*i
x8 = 8.11021064103792e-36 + 4.3942967040245*i
x9 = 62.8318530717959 - 4.3942967040245*i
x10 = 37.6991118430775 - 4.3942967040245*i
x11 = -94.2477796076938 - 4.3942967040245*i
x12 = -56.5486677646163 - 4.3942967040245*i
x13 = 50.2654824574367 + 4.3942967040245*i
x14 = -43.9822971502571 - 4.3942967040245*i
x15 = -37.6991118430775 + 4.3942967040245*i
x16 = -6.28318530717959 + 4.3942967040245*i
x17 = -94.2477796076938 + 4.3942967040245*i
x18 = 56.5486677646163 - 4.3942967040245*i
x19 = -81.6814089933346 + 4.3942967040245*i
x20 = -43.9822971502571 + 4.3942967040245*i
x21 = -12.5663706143592 + 4.3942967040245*i
x22 = 69.1150383789755 - 4.3942967040245*i
x23 = -18.8495559215388 + 4.3942967040245*i
x24 = -25.1327412287183 - 4.3942967040245*i
x25 = 43.9822971502571 + 4.3942967040245*i
x26 = -1514.24765903028 + 4.3942967040245*i
x27 = 1225.22113490002 + 4.3942967040245*i
x28 = 81.6814089933346 + 4.3942967040245*i
x29 = 25.1327412287183 + 4.3942967040245*i
x30 = 100.530964914873 + 4.3942967040245*i
x31 = 50.2654824574367 - 4.3942967040245*i
x32 = -125.663706143592 + 4.3942967040245*i
x33 = 6.28318530717959 + 4.3942967040245*i
x34 = 94.2477796076938 - 4.3942967040245*i
x35 = -37.6991118430775 - 4.3942967040245*i
x36 = 31.4159265358979 + 4.3942967040245*i
x37 = -81.6814089933346 - 4.3942967040245*i
x38 = -50.2654824574367 - 4.3942967040245*i
x39 = -31.4159265358979 - 4.3942967040245*i
x40 = 100.530964914873 - 4.3942967040245*i
x41 = -75.398223686155 - 4.3942967040245*i
x42 = -50.2654824574367 + 4.3942967040245*i
x43 = 18.8495559215388 + 4.3942967040245*i
x44 = -31.4159265358979 + 4.3942967040245*i
x45 = 2.9982965206586e-32 - 4.3942967040245*i
x46 = 94.2477796076938 + 4.3942967040245*i
x47 = 3166.72539481851 - 4.3942967040245*i
x48 = 18.8495559215388 - 4.3942967040245*i
x49 = 12.5663706143592 - 4.3942967040245*i
x50 = -18.8495559215388 - 4.3942967040245*i
x51 = -69.1150383789755 + 4.3942967040245*i
x52 = -100.530964914873 + 4.3942967040245*i
x53 = 6.28318530717959 - 4.3942967040245*i
x54 = 62.8318530717959 + 4.3942967040245*i
x55 = -11755.839709733 + 4.3942967040245*i
x56 = 25.1327412287183 - 4.3942967040245*i
x57 = 37.6991118430775 + 4.3942967040245*i
x58 = 7554636800.10092 - 4.3942967040245*i
x59 = -87.9645943005142 - 4.3942967040245*i
x60 = 43.9822971502571 - 4.3942967040245*i
x61 = -69.1150383789755 - 4.3942967040245*i
x62 = -62.8318530717959 - 4.3942967040245*i
x63 = -25.1327412287183 + 4.3942967040245*i
x64 = -75.398223686155 + 4.3942967040245*i
x65 = 69.1150383789755 + 4.3942967040245*i
x66 = -87.9645943005142 + 4.3942967040245*i
x67 = 169.646003293849 - 4.3942967040245*i
x68 = 12.5663706143592 + 4.3942967040245*i
x69 = -62.8318530717959 + 4.3942967040245*i
x70 = 81.6814089933346 - 4.3942967040245*i
x71 = -6.28318530717959 - 4.3942967040245*i
x72 = 87.9645943005142 - 4.3942967040245*i
x72 = 87.9645943005142 - 4.3942967040245*i
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