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- Thread starter bitrex
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CRGreathouse

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log(x)/x = e^0.1 is not the same as log(x) = x^0.1.For example, log(x) crosses the function x^0.1 somewhere less than e^e, and then falls below it again somewhere in the quadrillions. I'm wondering how one would go about solving the equation to find the Y axis intercepts? I want to find where log(x) = x^0.1, in trying to simplify the problem I end up with log(x)/x = e^0.1, but I'm not able to go any farther. The equation obviously has two solutions, so I imagine there's a polynomial involved somewhere? Any advice would be appreciated.

Generally, equations like this have no easy solutions (though they can be solved in terms of a special function called Lambert's W). The best way is probably by numerical methods:

Code:

```
gp> solve(x=1,9,log(x)-x^.1)
time = 0 ms.
%1 = 3.0597266796208088546065494702258610157
gp> solve(x=1e15,1e16,log(x)-x^.1)
time = 0 ms.
%2 = 3430631121407801.2027753365093892641824
```

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Mentallic

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exact solution is not possible in terms of your standard schoolbook functions

modern CASs like Maple will solve this using the Lambert W function...

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