Solving $\sin(x) + \ln(x) = 0$ without a calculator

Question

How would you algebraically handle

$$\sin(x) + \ln(x) = 0$$

to find the zeroes without a graphing calculator?

Here's the graph

Is approximation your only hope? Can you get an exact symbolic answer?

Answer 1

Finding the zero algebraically without a graphing calculator is possible, but a symbolic answer is impossible (I believe), so only the algebraic part can be answered.

There are multiple ways of doing it. The most straightforward one to me is to cast it as a fixed-point iteration $x\to e^{-\sin(x)}$. The Mathematica command NestList[Exp[-Sin[#]] &, 0, 20] gives a sequence of numbers converging towards 0.57871. The first four items of the sequence are

$$ 1,e^{-\sin (1)},e^{-\sin \left(e^{-\sin (1)}\right)},e^{-\sin \left(e^{-\sin \left(e^{-\sin (1)}\right)}\right)} $$

Certainly, not all fixed-point iterations converge, but in this case, it does.