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.

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:
$$W(1)=W\left(\frac{1}{x}e^\frac{1}{x}\right)$$
$$W(1)=\frac{1}{x}$$
$$x=\frac{1}{W(1)}$$
