Question

how can one obtain coordinates of intersections of two line diagrams with given expression or equation?

for example:
L1= sin(2x) , L2= Ln(x); or anything else.

Answer 1

Amazingly, nobody has yet suggested using the function designed to do this in matlab. Use fzero here. Fzero is a better choice than fsolve anyway, which requires the optimization toolbox. And, yes, you could do this with Newton’s method, or even bisection or the secant method. But reinventing the wheel is the wrong thing to do in general. Use functionality that already exists when it is there.

The problem at hand is to find a point where

sin(2*x) == log(x)

Here log(x) refers to the natural log. Do this by subtracting one from the other, then looking for a zero of the result.

fun = @(x) sin(2*x) - log(x);

Before you do so, ALWAYS plot it. ezplot can do that for you.

ezplot(fun)

The plot will show a single root that lies between 1 and 2.

fzero(fun,2)
ans =
       1.3994