Solution to x = 100 e − x / 100 ?How do you solve x = 100 e − x / 100 If I use ln then ln ⁡ ( x / 100 ) = − x / 100. How do I get from that to x = 56.7?

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Solution to   x  =  100      e          −      x              /            100      ?How do you solve  x  =  100      e          −      x              /            100      If I use   ln then   ln  ⁡  (  x      /    100  )  =  −  x      /    100. How do I get from that to   x  =  56.7?

Kali Galloway · Accepted Answer

Not solvable in a closed form using algebraic functions. It is a typical and well-known trascendental equation that has as a solution a special function called the Lambert W function.This function is defined as a principal branch of the solution of  x      e          x        =  y  →  x  =  W  (  y  )In your case, this is almost directly what you have. Rearrange:      x    100        e          x              /            100        =  1giving  x  =  100  W  (  1  )You need to solve for it numerically, or call ProductLog in Mathematica, which implements this function.A numerical solution is pretty straight-forward with a calculator. Write   y  =  x      /    100 and now you have  y  =      e          −      y      Start somewhere reasonable (like   y  =  1) and repeatedly plug it into this expression. You get a reasonable approximation in a few iterations:   1  ,  0.3678  ,  0.6922  ,  0.5004  ,  0.6062  ,  0.5453  ,  0.5796  ,  0.5601  ,  0.5711  ,  0.5648  ,  0.5684  ,  0.5664  ,  0.5675  ,  …

Almintas2l · Accepted Answer

If you want to solve numerically  f  (  y  )  =  y  −      e          −      y        =  0you can use very fast numerical methods.For example, Newton iterations will write      y          n      e      w        =            y      +      1                      e                  y                    +      1      Starting at   y  =  1 will provide the following successive iterates :   0.537883,   0.566987,   0.567143 which is the solution.As another example, Halley iterations will write      y          n      e      w        =            2      y      +      2              e        y            (      y      +      2      )      +              e                  2          y                    (      y      (      y      +      2      )      +      2      )      +      3              2                        (                      e            y                    +          1          )                3            Starting at   y  =  1 will provide the following successive iterates :   0.566599,   0.567143 which is the solution.

Solution to x=100e^(-x/100)?

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