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Ecuación x^2+5x=36
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Expresiones idénticas
- 2lnx+3lny
- 2lnx más 3lny
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- 2lnx-3lny
- 2*lnx+3*lny=1
2lnx+3lny la ecuación
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Solución
Ha introducido
[src]
2*log(x) + 3*log(y) = 0
$$2 \log{\left(x \right)} + 3 \log{\left(y \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$2 \log{\left(x \right)} + 3 \log{\left(y \right)} = 0$$
$$3 \log{\left(y \right)} = - 2 \log{\left(x \right)}$$
Devidimos ambás partes de la ecuación por el multiplicador de log =3
$$\log{\left(y \right)} = - \frac{2 \log{\left(x \right)}}{3}$$
Es la ecuación de la forma:
Por definición log
entonces
$$y = e^{\frac{\left(-1\right) 2 \log{\left(x \right)}}{3}}$$
simplificamos
$$y = \frac{1}{x^{\frac{2}{3}}}$$
$$2 \log{\left(x \right)} + 3 \log{\left(y \right)} = 0$$
$$3 \log{\left(y \right)} = - 2 \log{\left(x \right)}$$
Devidimos ambás partes de la ecuación por el multiplicador de log =3
$$\log{\left(y \right)} = - \frac{2 \log{\left(x \right)}}{3}$$
Es la ecuación de la forma:
log(v)=p
Por definición log
v=e^p
entonces
$$y = e^{\frac{\left(-1\right) 2 \log{\left(x \right)}}{3}}$$
simplificamos
$$y = \frac{1}{x^{\frac{2}{3}}}$$
Gráfica
Respuesta rápida
[src]
/2*atan2(im(x), re(x))\ /2*atan2(im(x), re(x))\
cos|---------------------| I*sin|---------------------|
\ 3 / \ 3 /
y1 = -------------------------- - ----------------------------
_________________ _________________
3 / 2 2 3 / 2 2
\/ im (x) + re (x) \/ im (x) + re (x)
$$y_{1} = - \frac{i \sin{\left(\frac{2 \operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{\sqrt[3]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}} + \frac{\cos{\left(\frac{2 \operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{\sqrt[3]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}}$$
y1 = -i*sin(2*atan2(im(x, re(x))/3)/(re(x)^2 + im(x)^2)^(1/3) + cos(2*atan2(im(x), re(x))/3)/(re(x)^2 + im(x)^2)^(1/3))
Suma y producto de raíces
[src]
suma
/2*atan2(im(x), re(x))\ /2*atan2(im(x), re(x))\
cos|---------------------| I*sin|---------------------|
\ 3 / \ 3 /
-------------------------- - ----------------------------
_________________ _________________
3 / 2 2 3 / 2 2
\/ im (x) + re (x) \/ im (x) + re (x)
$$- \frac{i \sin{\left(\frac{2 \operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{\sqrt[3]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}} + \frac{\cos{\left(\frac{2 \operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{\sqrt[3]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}}$$
=
/2*atan2(im(x), re(x))\ /2*atan2(im(x), re(x))\
cos|---------------------| I*sin|---------------------|
\ 3 / \ 3 /
-------------------------- - ----------------------------
_________________ _________________
3 / 2 2 3 / 2 2
\/ im (x) + re (x) \/ im (x) + re (x)
$$- \frac{i \sin{\left(\frac{2 \operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{\sqrt[3]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}} + \frac{\cos{\left(\frac{2 \operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{\sqrt[3]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}}$$
producto
/2*atan2(im(x), re(x))\ /2*atan2(im(x), re(x))\
cos|---------------------| I*sin|---------------------|
\ 3 / \ 3 /
-------------------------- - ----------------------------
_________________ _________________
3 / 2 2 3 / 2 2
\/ im (x) + re (x) \/ im (x) + re (x)
$$- \frac{i \sin{\left(\frac{2 \operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{\sqrt[3]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}} + \frac{\cos{\left(\frac{2 \operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{\sqrt[3]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}}$$
=
-2*I*atan2(im(x), re(x))
------------------------
3
e
-------------------------
_________________
3 / 2 2
\/ im (x) + re (x)
$$\frac{e^{- \frac{2 i \operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3}}}{\sqrt[3]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}}$$
exp(-2*i*atan2(im(x), re(x))/3)/(im(x)^2 + re(x)^2)^(1/3)