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log(x)-3+1/x=0

log(x)-3+1/x=0 equation

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Numerical solution:

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The solution

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             1    
log(x) - 3 + - = 0
             x
$$\left(\log{\left(x \right)} - 3\right) + \frac{1}{x} = 0$$
Simplify
The graph
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
           /  -3\
      3 + W\-e  /
x1 = e
$$x_{1} = e^{W\left(- \frac{1}{e^{3}}\right) + 3}$$ 
           /  -3    \
      3 + W\-e  , -1/
x2 = e
$$x_{2} = e^{W_{-1}\left(- \frac{1}{e^{3}}\right) + 3}$$ 
x2 = exp(LambertW(-exp(-3, -1) + 3))
Sum and product of roots [src] 
sum
      /  -3\         /  -3    \
 3 + W\-e  /    3 + W\-e  , -1/
e            + e
$$e^{W_{-1}\left(- \frac{1}{e^{3}}\right) + 3} + e^{W\left(- \frac{1}{e^{3}}\right) + 3}$$ 
=
      /  -3\         /  -3    \
 3 + W\-e  /    3 + W\-e  , -1/
e            + e
$$e^{W_{-1}\left(- \frac{1}{e^{3}}\right) + 3} + e^{W\left(- \frac{1}{e^{3}}\right) + 3}$$ 
product
      /  -3\       /  -3    \
 3 + W\-e  /  3 + W\-e  , -1/
e           *e
$$\frac{e^{W\left(- \frac{1}{e^{3}}\right) + 3}}{e^{-3 - W_{-1}\left(- \frac{1}{e^{3}}\right)}}$$ 
=
      /  -3\    /  -3    \
 6 + W\-e  / + W\-e  , -1/
e
$$e^{W_{-1}\left(- \frac{1}{e^{3}}\right) + W\left(- \frac{1}{e^{3}}\right) + 6}$$ 
exp(6 + LambertW(-exp(-3)) + LambertW(-exp(-3), -1))
Numerical answer [src] 
x1 = 19.0588374577227
x2 = 19.0588374577227 + 3.83592515622839e-18*i
x2 = 19.0588374577227 + 3.83592515622839e-18*i
The graph
log(x)-3+1/x=0 equation
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