Question

How to solve? ln(x)=2/x-2

Answer 1

Consider properties of the two sides of the function and deduce the number of possible roots

Explanation:

As this equation has the variable both inside and outside of the logarithm, it does not have an analytic solution in terms of simple functions. So we must deduce properties rather than simply solve.

Consider the behaviours of the two sides of the equation along the real line. `lnx` gives complex values for negative `x` while `2/x-2` gives real values, so there will be no solutions to the equation for `x<0`. `x=0` is a vertical asymptote for both sides of the equation - so while it might in some situations be useful to regard it as a solution, it isn't common to do so.

For positive `x`, `ln x` increases monotonically from `-oo` to `+oo`, while `2/x-2` decreases monotonically from `+oo` to `-2` (compare the graph of `1/x`). So these curves cross over once and once only along the real line - the equation has one real solution.

In general, solving this type of problem requires numerical approximation. However, in this particular example, trialling some simple values immediately delivers us the answer: `ln(1)=0` and `2/1-2=0`. So `x=1` is the one solution that we seek.

Sanity check this by overplotting the two sides of the equation and observing where they cross:
graph{(y-ln(x))(y-(2/x-2))=0 [-10, 10, -5, 5]}