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

1. Jul 25, 2010

deltapapazulu

1. The problem statement, all variables and given/known data

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'.

2. Relevant equations

3. The attempt at a solution

2. Jul 25, 2010

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.

3. Jul 25, 2010

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.

4. Jul 25, 2010

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 .

5. Jul 25, 2010

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

Reference:
http://en.wikipedia.org/wiki/Lambert_W" [Broken]
http://reference.wolfram.com/mathematica/ref/ProductLog.html" [Broken]

Last edited by a moderator: May 4, 2017