Ln(t) - t = 2; Solve for 't'?

[deltapapazulu]

Homework Statement

Is there a simple way to solve for 't' in the equation: ln(t) - t = 2 ? Just a curiosity. Not for a class. I've browsed to texts that cover logarithmic equations and haven't found a single problem or rule say how to solve for 't'.

Homework Equations

The Attempt at a Solution

 

[rock.freak667]

There is no way to get 't' in terms of elementary functions, but you can solve it graphically or perhaps using another complex function.

 

[ajny]

Trancendental equation.
No closed form solution.
Turn it into a game.
Calculate ln(t)-t vs t in Excel.
Play with t.
Stop when you get to an answer accurate to 6 decimal places, ie. 2.000001.
Now you will have developed some feel for the iterative solution of a trancendental equation. You can then read more about Newton-Raphson, etc.

 

[Mute]

Exponentiate both sides to get:

$$te^{-t} = e^2$$

Now, multiply both sides by -1 to get $-te^{-t} = -e^2$. This is now of the form of the Lambert-W function, defined by

$$W(z)e^{W(z)} = z,$$

which determines t in terms of the lambert W function. This function is not an elementary function, but much is known about it. See http://en.wikipedia.org/wiki/Lambert_W .

 

[Orion1]

Mathematica 6 found a solution, however the numerical solution has a imaginary axis component.

ln(t) - t = 2

t = - ProductLog[-e^2]

ProductLog[z]
gives the principal solution for w in z=we^w.

The ProductLog[z] function is the Lambert-W function, however Mathematica 6 and 7 documentation does not mention this.

t = -1.13902 - 2.07318i
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Reference:
http://en.wikipedia.org/wiki/Lambert_W"
http://reference.wolfram.com/mathematica/ref/ProductLog.html"