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exp(20*x)=113/252 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 20*x   113
e     = ---
        252
e20x=113252e^{20 x} = \frac{113}{252}
Simplificar
Solución detallada
Tenemos la ecuación:
e20x=113252e^{20 x} = \frac{113}{252}
o
e20x−113252=0e^{20 x} - \frac{113}{252} = 0
o
e20x=113252e^{20 x} = \frac{113}{252}
o
e20x=113252e^{20 x} = \frac{113}{252}
- es la ecuación exponencial más simple
Sustituimos
v=e20xv = e^{20 x}
obtendremos
v−113252=0v - \frac{113}{252} = 0
o
v−113252=0v - \frac{113}{252} = 0
Transportamos los términos libres (sin v)
del miembro izquierdo al derecho, obtenemos:
v=113252v = \frac{113}{252}
Obtenemos la respuesta: v = 113/252
hacemos cambio inverso
e20x=ve^{20 x} = v
o
x=log⁡(v)20x = \frac{\log{\left(v \right)}}{20}
Entonces la respuesta definitiva es
x1=log⁡(113252)log⁡(e20)=−log⁡(252)20+log⁡(113)20x_{1} = \frac{\log{\left(\frac{113}{252} \right)}}{\log{\left(e^{20} \right)}} = - \frac{\log{\left(252 \right)}}{20} + \frac{\log{\left(113 \right)}}{20}
Gráfica
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.505e86
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
        /       19        \
        |       --        |
        | 9/10  20 20_____|
        |6    *7  *\/ 113 |
x1 = log|-----------------|
        \        42       /
x1=log⁡(11320⋅6910⋅7192042)x_{1} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} 
                   /       19        \
                   |       --        |
                   | 9/10  20 20_____|
       9*pi*I      |6    *7  *\/ 113 |
x2 = - ------ + log|-----------------|
         10        \        42       /
x2=log⁡(11320⋅6910⋅7192042)−9iπ10x_{2} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} - \frac{9 i \pi}{10} 
                   /       19        \
                   |       --        |
                   | 9/10  20 20_____|
       4*pi*I      |6    *7  *\/ 113 |
x3 = - ------ + log|-----------------|
         5         \        42       /
x3=log⁡(11320⋅6910⋅7192042)−4iπ5x_{3} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} - \frac{4 i \pi}{5} 
                   /       19        \
                   |       --        |
                   | 9/10  20 20_____|
       7*pi*I      |6    *7  *\/ 113 |
x4 = - ------ + log|-----------------|
         10        \        42       /
x4=log⁡(11320⋅6910⋅7192042)−7iπ10x_{4} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} - \frac{7 i \pi}{10} 
                   /       19        \
                   |       --        |
                   | 9/10  20 20_____|
       3*pi*I      |6    *7  *\/ 113 |
x5 = - ------ + log|-----------------|
         5         \        42       /
x5=log⁡(11320⋅6910⋅7192042)−3iπ5x_{5} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} - \frac{3 i \pi}{5} 
                 /       19        \
                 |       --        |
                 | 9/10  20 20_____|
       pi*I      |6    *7  *\/ 113 |
x6 = - ---- + log|-----------------|
        2        \        42       /
x6=log⁡(11320⋅6910⋅7192042)−iπ2x_{6} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} - \frac{i \pi}{2} 
                   /       19        \
                   |       --        |
                   | 9/10  20 20_____|
       2*pi*I      |6    *7  *\/ 113 |
x7 = - ------ + log|-----------------|
         5         \        42       /
x7=log⁡(11320⋅6910⋅7192042)−2iπ5x_{7} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} - \frac{2 i \pi}{5} 
                   /       19        \
                   |       --        |
                   | 9/10  20 20_____|
       3*pi*I      |6    *7  *\/ 113 |
x8 = - ------ + log|-----------------|
         10        \        42       /
x8=log⁡(11320⋅6910⋅7192042)−3iπ10x_{8} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} - \frac{3 i \pi}{10} 
                 /       19        \
                 |       --        |
                 | 9/10  20 20_____|
       pi*I      |6    *7  *\/ 113 |
x9 = - ---- + log|-----------------|
        5        \        42       /
x9=log⁡(11320⋅6910⋅7192042)−iπ5x_{9} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} - \frac{i \pi}{5} 
                  /       19        \
                  |       --        |
                  | 9/10  20 20_____|
        pi*I      |6    *7  *\/ 113 |
x10 = - ---- + log|-----------------|
         10       \        42       /
x10=log⁡(11320⋅6910⋅7192042)−iπ10x_{10} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} - \frac{i \pi}{10} 
                /       19        \
                |       --        |
                | 9/10  20 20_____|
      pi*I      |6    *7  *\/ 113 |
x11 = ---- + log|-----------------|
       10       \        42       /
x11=log⁡(11320⋅6910⋅7192042)+iπ10x_{11} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} + \frac{i \pi}{10} 
                /       19        \
                |       --        |
                | 9/10  20 20_____|
      pi*I      |6    *7  *\/ 113 |
x12 = ---- + log|-----------------|
       5        \        42       /
x12=log⁡(11320⋅6910⋅7192042)+iπ5x_{12} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} + \frac{i \pi}{5} 
                  /       19        \
                  |       --        |
                  | 9/10  20 20_____|
      3*pi*I      |6    *7  *\/ 113 |
x13 = ------ + log|-----------------|
        10        \        42       /
x13=log⁡(11320⋅6910⋅7192042)+3iπ10x_{13} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} + \frac{3 i \pi}{10} 
                  /       19        \
                  |       --        |
                  | 9/10  20 20_____|
      2*pi*I      |6    *7  *\/ 113 |
x14 = ------ + log|-----------------|
        5         \        42       /
x14=log⁡(11320⋅6910⋅7192042)+2iπ5x_{14} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} + \frac{2 i \pi}{5} 
                /       19        \
                |       --        |
                | 9/10  20 20_____|
      pi*I      |6    *7  *\/ 113 |
x15 = ---- + log|-----------------|
       2        \        42       /
x15=log⁡(11320⋅6910⋅7192042)+iπ2x_{15} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} + \frac{i \pi}{2} 
                  /       19        \
                  |       --        |
                  | 9/10  20 20_____|
      3*pi*I      |6    *7  *\/ 113 |
x16 = ------ + log|-----------------|
        5         \        42       /
x16=log⁡(11320⋅6910⋅7192042)+3iπ5x_{16} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} + \frac{3 i \pi}{5} 
                  /       19        \
                  |       --        |
                  | 9/10  20 20_____|
      7*pi*I      |6    *7  *\/ 113 |
x17 = ------ + log|-----------------|
        10        \        42       /
x17=log⁡(11320⋅6910⋅7192042)+7iπ10x_{17} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} + \frac{7 i \pi}{10} 
                  /       19        \
                  |       --        |
                  | 9/10  20 20_____|
      4*pi*I      |6    *7  *\/ 113 |
x18 = ------ + log|-----------------|
        5         \        42       /
x18=log⁡(11320⋅6910⋅7192042)+4iπ5x_{18} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} + \frac{4 i \pi}{5} 
                  /       19        \
                  |       --        |
                  | 9/10  20 20_____|
      9*pi*I      |6    *7  *\/ 113 |
x19 = ------ + log|-----------------|
        10        \        42       /
x19=log⁡(11320⋅6910⋅7192042)+9iπ10x_{19} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} + \frac{9 i \pi}{10} 
                /       19        \
                |       --        |
                | 9/10  20 20_____|
                |6    *7  *\/ 113 |
x20 = pi*I + log|-----------------|
                \        42       /
x20=log⁡(11320⋅6910⋅7192042)+iπx_{20} = \log{\left(\frac{\sqrt[20]{113} \cdot 6^{\frac{9}{10}} \cdot 7^{\frac{19}{20}}}{42} \right)} + i \pi 
x20 = log(113^(1/20)*6^(9/10)*7^(19/20)/42) + i*pi
Respuesta numérica [src] 
x1 = -0.0401020634399541 - 2.82743338823081*i
x2 = -0.0401020634399541 - 2.51327412287183*i
x3 = -0.0401020634399541 - 2.19911485751286*i
x4 = -0.0401020634399541 - 1.88495559215388*i
x5 = -0.0401020634399541 - 1.5707963267949*i
x6 = -0.0401020634399541 - 1.25663706143592*i
x7 = -0.0401020634399541 - 0.942477796076938*i
x8 = -0.0401020634399541 - 0.628318530717959*i
x9 = -0.0401020634399541 - 0.314159265358979*i
x10 = -0.0401020634399541 + 0.314159265358979*i
x11 = -0.0401020634399541 + 0.628318530717959*i
x12 = -0.0401020634399541 + 0.942477796076938*i
x13 = -0.0401020634399541 + 1.25663706143592*i
x14 = -0.0401020634399541 + 1.5707963267949*i
x15 = -0.0401020634399541 + 1.88495559215388*i
x16 = -0.0401020634399541 + 2.19911485751286*i
x17 = -0.0401020634399541 + 2.51327412287183*i
x18 = -0.0401020634399541 + 2.82743338823081*i
x19 = -0.0401020634399541 + 3.14159265358979*i
x20 = -0.0401020634399541
x20 = -0.0401020634399541
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