Getting stuck on simple logarithmic equation: $x \times \ln (x) = 1$ $$x \times \ln (x) = 1$$
I am trying to solve that equation. I used the theory $\ln(a) = \ln(b)$ being equivalent to $a = b$ and got stuck at
$$x = e^{\frac{1}{x}}$$
That's as far as I went and I know there's a solution (around 1.8 or 1.9), since I used my calculator, but I'd like to know how to do this by hand.
 A: You can use the law of logarithms which states that for $a,b\in\mathbb{R}$: $a\ln{b}=\ln\left(b^{a}\right)$.
Therefore, you have:
$$x\ln{x}=1 \implies \ln{x^{x}}=1$$
You hence have:
$$x^{x}=e$$
Which does not have an elementary closed form, so you must use numerical methods (for instance Newton-Raphson iteration) to get an approximation (Mathematica gives $x\approx 1.76322$).
If you're interested, the closed form solution is: $$\frac{1}{W(1)}, \qquad \text{ where } W(z) \text{ is the LambertW function}$$
A: There is no solution using only algebraic manipulation.  We have to use the "product-log" or Lambert-W function to solve this, and this function doesn't fall in the "simple functions" category. :)
Basically, the Lambert-W function is the inverse function of:
$$f(x) = xe^x$$
Equivalently:
$$x=W(xe^x)$$
So, using your expression:
$$x=e^\frac{1}{x}$$
$$1=\frac{1}{x}e^\frac{1}{x}$$
Taking the product-log of both sides:
$$W(1)=W\left(\frac{1}{x}e^\frac{1}{x}\right)$$
$$W(1)=\frac{1}{x}$$
$$x=\frac{1}{W(1)}$$
A: This equation won't have an elementary solution - you'll have to solve it numerically.  (Or you can ask WA).
A: It's in fact impossible to express your variable $x$ as a combination of usual elementary function. Symbolically, you shall express it using Lambert's L function, which solves the equation
$$x=L(x)e^{L(x)}$$
For instance, if $$xe^x = 1$$ then $x=L(1)$ as is easily seen. By expressing $x$ as a the logarithm of some other number $y$, one has from the preceding equation
$$y\ln y=1$$
Then, the solution of your equation would be $\ln(y)=L(1)$ or $y=e^{L(1)}$. Take a look at wikipedia's http://en.wikipedia.org/wiki/Lambert_W_function for further information!
A: ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$See Example 4 here: http://en.wikipedia.org/wiki/Lambert_W_function
