log(x)=1/k (уравнение)

Найду корень уравнения: log(x)=1/k

Вы ввели

         1
log(x) = -
         k

$$\log{\left(x \right)} = \frac{1}{k}$$

Подробное решение

Дано уравнение
$$\log{\left(x \right)} = \frac{1}{k}$$
$$\log{\left(x \right)} = \frac{1}{k}$$
Это уравнение вида:

log(v)=p

По определению log

v=e^p

тогда
$$x = e^{\frac{1}{k}}$$
упрощаем
$$x = e^{\frac{1}{k}}$$

Быстрый ответ

                                re(k)                re(k)                          
                           ---------------      ---------------                     
                             2        2           2        2                        
        /     im(k)     \  im (k) + re (k)      im (k) + re (k)    /     im(k)     \
x1 = cos|---------------|*e                - I*e               *sin|---------------|
        |  2        2   |                                          |  2        2   |
        \im (k) + re (k)/                                          \im (k) + re (k)/

$$x_{1} = - i e^{\frac{\operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}} \sin{\left(\frac{\operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} \right)} + e^{\frac{\operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}} \cos{\left(\frac{\operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} \right)}$$

Сумма и произведение корней

сумма

                           re(k)                re(k)                          
                      ---------------      ---------------                     
                        2        2           2        2                        
   /     im(k)     \  im (k) + re (k)      im (k) + re (k)    /     im(k)     \
cos|---------------|*e                - I*e               *sin|---------------|
   |  2        2   |                                          |  2        2   |
   \im (k) + re (k)/                                          \im (k) + re (k)/

$$- i e^{\frac{\operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}} \sin{\left(\frac{\operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} \right)} + e^{\frac{\operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}} \cos{\left(\frac{\operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} \right)}$$

=

                           re(k)                re(k)                          
                      ---------------      ---------------                     
                        2        2           2        2                        
   /     im(k)     \  im (k) + re (k)      im (k) + re (k)    /     im(k)     \
cos|---------------|*e                - I*e               *sin|---------------|
   |  2        2   |                                          |  2        2   |
   \im (k) + re (k)/                                          \im (k) + re (k)/

$$- i e^{\frac{\operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}} \sin{\left(\frac{\operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} \right)} + e^{\frac{\operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}} \cos{\left(\frac{\operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} \right)}$$

произведение

                           re(k)                re(k)                          
                      ---------------      ---------------                     
                        2        2           2        2                        
   /     im(k)     \  im (k) + re (k)      im (k) + re (k)    /     im(k)     \
cos|---------------|*e                - I*e               *sin|---------------|
   |  2        2   |                                          |  2        2   |
   \im (k) + re (k)/                                          \im (k) + re (k)/

$$- i e^{\frac{\operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}} \sin{\left(\frac{\operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} \right)} + e^{\frac{\operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}} \cos{\left(\frac{\operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} \right)}$$

=

      re(k)            I*im(k)    
 --------------- - ---------------
   2        2        2        2   
 im (k) + re (k)   im (k) + re (k)
e                                 

$$e^{\frac{\operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} - \frac{i \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}}$$