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(0,2)^x-2=5^x2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 -x        x2
5   - 2 = 5
−2+(15)x=5x2-2 + \left(\frac{1}{5}\right)^{x} = 5^{x_{2}}
Simplificar
Solución detallada
Tenemos la ecuación:
−2+(15)x=5x2-2 + \left(\frac{1}{5}\right)^{x} = 5^{x_{2}}
o
−5x2+(−2+(15)x)=0- 5^{x_{2}} + \left(-2 + \left(\frac{1}{5}\right)^{x}\right) = 0
Sustituimos
v=(15)xv = \left(\frac{1}{5}\right)^{x}
obtendremos
−5x2+v−2=0- 5^{x_{2}} + v - 2 = 0
o
−5x2+v−2=0- 5^{x_{2}} + v - 2 = 0
hacemos cambio inverso
(15)x=v\left(\frac{1}{5}\right)^{x} = v
o
x=−log⁡(v)log⁡(5)x = - \frac{\log{\left(v \right)}}{\log{\left(5 \right)}}
Entonces la respuesta definitiva es
x1=log⁡(log⁡(15x2+2)log⁡(5))log⁡(15)=−log⁡(log⁡(15x2+2)log⁡(5))log⁡(5)x_{1} = \frac{\log{\left(\frac{\log{\left(\frac{1}{5^{x_{2}} + 2} \right)}}{\log{\left(5 \right)}} \right)}}{\log{\left(\frac{1}{5} \right)}} = - \frac{\log{\left(\frac{\log{\left(\frac{1}{5^{x_{2}} + 2} \right)}}{\log{\left(5 \right)}} \right)}}{\log{\left(5 \right)}}
Gráfica
Respuesta rápida [src] 
        /    1    \        /   1   \
     log|---------|   I*arg|-------|
        ||     x2||        |     x2|
        \|2 + 5  |/        \2 + 5  /
x1 = -------------- + --------------
         log(5)           log(5)
x1=log⁡(1∣5x2+2∣)log⁡(5)+iarg⁡(15x2+2)log⁡(5)x_{1} = \frac{\log{\left(\frac{1}{\left|{5^{x_{2}} + 2}\right|} \right)}}{\log{\left(5 \right)}} + \frac{i \arg{\left(\frac{1}{5^{x_{2}} + 2} \right)}}{\log{\left(5 \right)}} 
x1 = log(1/|5^x2 + 2|)/log(5) + i*arg(1/(5^x2 + 2))/log(5)
Suma y producto de raíces [src] 
suma
   /    1    \        /   1   \
log|---------|   I*arg|-------|
   ||     x2||        |     x2|
   \|2 + 5  |/        \2 + 5  /
-------------- + --------------
    log(5)           log(5)
log⁡(1∣5x2+2∣)log⁡(5)+iarg⁡(15x2+2)log⁡(5)\frac{\log{\left(\frac{1}{\left|{5^{x_{2}} + 2}\right|} \right)}}{\log{\left(5 \right)}} + \frac{i \arg{\left(\frac{1}{5^{x_{2}} + 2} \right)}}{\log{\left(5 \right)}} 
=
   /    1    \        /   1   \
log|---------|   I*arg|-------|
   ||     x2||        |     x2|
   \|2 + 5  |/        \2 + 5  /
-------------- + --------------
    log(5)           log(5)
log⁡(1∣5x2+2∣)log⁡(5)+iarg⁡(15x2+2)log⁡(5)\frac{\log{\left(\frac{1}{\left|{5^{x_{2}} + 2}\right|} \right)}}{\log{\left(5 \right)}} + \frac{i \arg{\left(\frac{1}{5^{x_{2}} + 2} \right)}}{\log{\left(5 \right)}} 
producto
   /    1    \        /   1   \
log|---------|   I*arg|-------|
   ||     x2||        |     x2|
   \|2 + 5  |/        \2 + 5  /
-------------- + --------------
    log(5)           log(5)
log⁡(1∣5x2+2∣)log⁡(5)+iarg⁡(15x2+2)log⁡(5)\frac{\log{\left(\frac{1}{\left|{5^{x_{2}} + 2}\right|} \right)}}{\log{\left(5 \right)}} + \frac{i \arg{\left(\frac{1}{5^{x_{2}} + 2} \right)}}{\log{\left(5 \right)}} 
=
     /   1   \      /    1    \
I*arg|-------| + log|---------|
     |     x2|      ||     x2||
     \2 + 5  /      \|2 + 5  |/
-------------------------------
             log(5)
log⁡(1∣5x2+2∣)+iarg⁡(15x2+2)log⁡(5)\frac{\log{\left(\frac{1}{\left|{5^{x_{2}} + 2}\right|} \right)} + i \arg{\left(\frac{1}{5^{x_{2}} + 2} \right)}}{\log{\left(5 \right)}} 
(i*arg(1/(2 + 5^x2)) + log(1/|2 + 5^x2|))/log(5)
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