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3log(4)^2-7*log(4)^x+2=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
     2           x           
3*log (4) - 7*log (4) + 2 = 0
$$\left(- 7 \log{\left(4 \right)}^{x} + 3 \log{\left(4 \right)}^{2}\right) + 2 = 0$$
Solución detallada
Tenemos la ecuación:
$$\left(- 7 \log{\left(4 \right)}^{x} + 3 \log{\left(4 \right)}^{2}\right) + 2 = 0$$
o
$$\left(- 7 \log{\left(4 \right)}^{x} + 3 \log{\left(4 \right)}^{2}\right) + 2 = 0$$
Sustituimos
$$v = 1$$
obtendremos
$$- 7 \log{\left(4 \right)}^{x} + 2 + 3 \log{\left(4 \right)}^{2} = 0$$
o
$$- 7 \log{\left(4 \right)}^{x} + 2 + 3 \log{\left(4 \right)}^{2} = 0$$
hacemos cambio inverso
$$1 = v$$
o
$$x = \tilde{\infty} \log{\left(v \right)}$$
Entonces la respuesta definitiva es
$$x_{1} = \frac{\log{\left(\log{\left(\left(\frac{2}{7} + \frac{3 \log{\left(4 \right)}^{2}}{7}\right)^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}} \right)} \right)}}{\log{\left(1 \right)}} = \tilde{\infty}$$
Gráfica
Suma y producto de raíces [src]
suma
   /                     1     \      /                    1     \
   |                -----------|      |               -----------|
   |                log(log(4))|      |               log(log(4))|
   |/          2   \           |      |/         2   \           |
   ||2   12*log (2)|           |      ||2   3*log (4)|           |
log||- + ----------|           | + log||- + ---------|           |
   \\7       7     /           /      \\7       7    /           /
$$\log{\left(\left(\frac{2}{7} + \frac{12 \log{\left(2 \right)}^{2}}{7}\right)^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}} \right)} + \log{\left(\left(\frac{2}{7} + \frac{3 \log{\left(4 \right)}^{2}}{7}\right)^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}} \right)}$$
=
   /                    1     \      /                     1     \
   |               -----------|      |                -----------|
   |               log(log(4))|      |                log(log(4))|
   |/         2   \           |      |/          2   \           |
   ||2   3*log (4)|           |      ||2   12*log (2)|           |
log||- + ---------|           | + log||- + ----------|           |
   \\7       7    /           /      \\7       7     /           /
$$\log{\left(\left(\frac{2}{7} + \frac{3 \log{\left(4 \right)}^{2}}{7}\right)^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}} \right)} + \log{\left(\left(\frac{2}{7} + \frac{12 \log{\left(2 \right)}^{2}}{7}\right)^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}} \right)}$$
producto
   /                     1     \    /                    1     \
   |                -----------|    |               -----------|
   |                log(log(4))|    |               log(log(4))|
   |/          2   \           |    |/         2   \           |
   ||2   12*log (2)|           |    ||2   3*log (4)|           |
log||- + ----------|           |*log||- + ---------|           |
   \\7       7     /           /    \\7       7    /           /
$$\log{\left(\left(\frac{2}{7} + \frac{12 \log{\left(2 \right)}^{2}}{7}\right)^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}} \right)} \log{\left(\left(\frac{2}{7} + \frac{3 \log{\left(4 \right)}^{2}}{7}\right)^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}} \right)}$$
=
   /                   /                                 1     \\
   |                   |     -1                     -----------||
   |                   | -----------                log(log(4))||
   |                   | log(log(4)) /         2   \           ||
   |                log\7           *\2 + 3*log (4)/           /|
   |                --------------------------------------------|
   |                            log(2) + log(log(2))            |
   |/          2   \                                            |
   ||2   12*log (2)|                                            |
log||- + ----------|                                            |
   \\7       7     /                                            /
$$\log{\left(\left(\frac{2}{7} + \frac{12 \log{\left(2 \right)}^{2}}{7}\right)^{\frac{\log{\left(\frac{\left(2 + 3 \log{\left(4 \right)}^{2}\right)^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}}}{7^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}}} \right)}}{\log{\left(\log{\left(2 \right)} \right)} + \log{\left(2 \right)}}} \right)}$$
log((2/7 + 12*log(2)^2/7)^(log(7^(-1/log(log(4)))*(2 + 3*log(4)^2)^(1/log(log(4))))/(log(2) + log(log(2)))))
Respuesta rápida [src]
        /                     1     \
        |                -----------|
        |                log(log(4))|
        |/          2   \           |
        ||2   12*log (2)|           |
x1 = log||- + ----------|           |
        \\7       7     /           /
$$x_{1} = \log{\left(\left(\frac{2}{7} + \frac{12 \log{\left(2 \right)}^{2}}{7}\right)^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}} \right)}$$
        /                    1     \
        |               -----------|
        |               log(log(4))|
        |/         2   \           |
        ||2   3*log (4)|           |
x2 = log||- + ---------|           |
        \\7       7    /           /
$$x_{2} = \log{\left(\left(\frac{2}{7} + \frac{3 \log{\left(4 \right)}^{2}}{7}\right)^{\frac{1}{\log{\left(\log{\left(4 \right)} \right)}}} \right)}$$
x2 = log((2/7 + 3*log(4)^2/7)^(1/log(log(4))))
Respuesta numérica [src]
x1 = 0.317702364939076
x2 = 0.317702364939076
x2 = 0.317702364939076