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- La ecuación:
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Ecuación x=(x+1)/2
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Ecuación 350-(45+a)=60
-
Ecuación x^x=x
-
Ecuación x^2+8*x-13=0
- Expresar {x} en función de y en la ecuación:
- 11*x-12*y=11
- 8*x-13*y=-12
- 19*x-2*y=-17
- -16*x-2*y=4
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Expresiones idénticas
- (uno / seis)^x+ uno = treinta y seis ^x- uno
- (1 dividir por 6) en el grado x más 1 es igual a 36 en el grado x menos 1
- (uno dividir por seis) en el grado x más uno es igual a treinta y seis en el grado x menos uno
- (1/6)x+1=36x-1
- 1/6x+1=36x-1
- 1/6^x+1=36^x-1
- (1 dividir por 6)^x+1=36^x-1
-
Expresiones semejantes
- (1/6)^x+1=36^x+1
- (1/6)^x-1=36^x-1
(1/6)^x+1=36^x-1 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación:
$$1 + \left(\frac{1}{6}\right)^{x} = 36^{x} - 1$$
o
$$\left(1 + \left(\frac{1}{6}\right)^{x}\right) + \left(1 - 36^{x}\right) = 0$$
Sustituimos
$$v = \left(\frac{1}{6}\right)^{x}$$
obtendremos
$$2 - \frac{1}{v^{2}} + 6^{- x} = 0$$
o
$$2 - \frac{1}{v^{2}} + 6^{- x} = 0$$
hacemos cambio inverso
$$\left(\frac{1}{6}\right)^{x} = v$$
o
$$x = - \frac{\log{\left(v \right)}}{\log{\left(6 \right)}}$$
Entonces la respuesta definitiva es
$$x_{1} = \frac{\log{\left(\frac{- \log{\left(2 \right)} + \log{\left(1 + \sqrt{5} \right)}}{\log{\left(6 \right)}} \right)}}{\log{\left(\frac{1}{6} \right)}} = \frac{\log{\left(\log{\left(6 \right)} \right)} - \log{\left(- \log{\left(2 \right)} + \log{\left(1 + \sqrt{5} \right)} \right)}}{\log{\left(6 \right)}}$$
$$x_{2} = \frac{\log{\left(\frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi}{\log{\left(6 \right)}} \right)}}{\log{\left(\frac{1}{6} \right)}} = - \frac{\log{\left(\frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi}{\log{\left(6 \right)}} \right)}}{\log{\left(6 \right)}}$$
$$x_{3} = \frac{\log{\left(\frac{i \pi}{\log{\left(6 \right)}} \right)}}{\log{\left(\frac{1}{6} \right)}} = - \frac{\log{\left(\frac{i \pi}{\log{\left(6 \right)}} \right)}}{\log{\left(6 \right)}}$$
$$1 + \left(\frac{1}{6}\right)^{x} = 36^{x} - 1$$
o
$$\left(1 + \left(\frac{1}{6}\right)^{x}\right) + \left(1 - 36^{x}\right) = 0$$
Sustituimos
$$v = \left(\frac{1}{6}\right)^{x}$$
obtendremos
$$2 - \frac{1}{v^{2}} + 6^{- x} = 0$$
o
$$2 - \frac{1}{v^{2}} + 6^{- x} = 0$$
hacemos cambio inverso
$$\left(\frac{1}{6}\right)^{x} = v$$
o
$$x = - \frac{\log{\left(v \right)}}{\log{\left(6 \right)}}$$
Entonces la respuesta definitiva es
$$x_{1} = \frac{\log{\left(\frac{- \log{\left(2 \right)} + \log{\left(1 + \sqrt{5} \right)}}{\log{\left(6 \right)}} \right)}}{\log{\left(\frac{1}{6} \right)}} = \frac{\log{\left(\log{\left(6 \right)} \right)} - \log{\left(- \log{\left(2 \right)} + \log{\left(1 + \sqrt{5} \right)} \right)}}{\log{\left(6 \right)}}$$
$$x_{2} = \frac{\log{\left(\frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi}{\log{\left(6 \right)}} \right)}}{\log{\left(\frac{1}{6} \right)}} = - \frac{\log{\left(\frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi}{\log{\left(6 \right)}} \right)}}{\log{\left(6 \right)}}$$
$$x_{3} = \frac{\log{\left(\frac{i \pi}{\log{\left(6 \right)}} \right)}}{\log{\left(\frac{1}{6} \right)}} = - \frac{\log{\left(\frac{i \pi}{\log{\left(6 \right)}} \right)}}{\log{\left(6 \right)}}$$
Suma y producto de raíces
[src]
suma
/ ___\
| 1 \/ 5 |
/ ___\ log|- - + -----|
-log(2) + log\1 + \/ 5 / \ 2 2 / pi*I pi*I
------------------------ + ---------------- + ------ + ------
log(6) log(6) log(6) log(6)
$$\frac{i \pi}{\log{\left(6 \right)}} + \left(\frac{- \log{\left(2 \right)} + \log{\left(1 + \sqrt{5} \right)}}{\log{\left(6 \right)}} + \left(\frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{\log{\left(6 \right)}} + \frac{i \pi}{\log{\left(6 \right)}}\right)\right)$$
=
/ ___\
| 1 \/ 5 |
/ ___\ log|- - + -----|
-log(2) + log\1 + \/ 5 / \ 2 2 / 2*pi*I
------------------------ + ---------------- + ------
log(6) log(6) log(6)
$$\frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{\log{\left(6 \right)}} + \frac{- \log{\left(2 \right)} + \log{\left(1 + \sqrt{5} \right)}}{\log{\left(6 \right)}} + \frac{2 i \pi}{\log{\left(6 \right)}}$$
producto
/ / ___\ \
| | 1 \/ 5 | |
/ ___\ |log|- - + -----| |
-log(2) + log\1 + \/ 5 / | \ 2 2 / pi*I | pi*I
------------------------*|---------------- + ------|*------
log(6) \ log(6) log(6)/ log(6)
$$\frac{- \log{\left(2 \right)} + \log{\left(1 + \sqrt{5} \right)}}{\log{\left(6 \right)}} \left(\frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{\log{\left(6 \right)}} + \frac{i \pi}{\log{\left(6 \right)}}\right) \frac{i \pi}{\log{\left(6 \right)}}$$
=
/ pi \
| -------|
| 3 |
/ / ___\\ | log (6)|
| | 1 \/ 5 || |/ 2 \ |
-I*|pi*I + log|- - + -----||*log||---------| |
\ \ 2 2 // || ___| |
\\1 + \/ 5 / /
$$- i \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) \log{\left(\left(\frac{2}{1 + \sqrt{5}}\right)^{\frac{\pi}{\log{\left(6 \right)}^{3}}} \right)}$$
-i*(pi*i + log(-1/2 + sqrt(5)/2))*log((2/(1 + sqrt(5)))^(pi/log(6)^3))
Respuesta rápida
[src]
/ ___\
-log(2) + log\1 + \/ 5 /
x1 = ------------------------
log(6)
$$x_{1} = \frac{- \log{\left(2 \right)} + \log{\left(1 + \sqrt{5} \right)}}{\log{\left(6 \right)}}$$
/ ___\
| 1 \/ 5 |
log|- - + -----|
\ 2 2 / pi*I
x2 = ---------------- + ------
log(6) log(6)
$$x_{2} = \frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{\log{\left(6 \right)}} + \frac{i \pi}{\log{\left(6 \right)}}$$
pi*I
x3 = ------
log(6)
$$x_{3} = \frac{i \pi}{\log{\left(6 \right)}}$$
x3 = i*pi/log(6)
Respuesta numérica
[src]
x1 = 0.268569433187884
x2 = -0.268569433187884 + 1.75335624426379*i
x3 = 1.75335624426379*i
x3 = 1.75335624426379*i