Whether or not you can solve it "analytically" depends upon what you mean by "analytically". From we get and the . From that, where "W" is the "Lambert W function" which is defined as the inverse function to . Of course, values of the W function are calculated numerically, just as values of logarithms, exponentials, or trig functions are.
"Linear Interpolation", that Sammy S referred to, is also called the "secant method" since you use a line between two points on the curve to approximate the curve rather than the tangent line as Newton's method does. Now that you know that (0.5, -0.9131) and (1, 1) are on the curve, you could calculate the straight line between those two points and and determine where that line crosses the y-axis. Take that together with either 0.5 or 1, depending upon whether the actual value of the function is positive or negative, and do it again.
If you don't have a computer doing the calculations for you "bisection" is simpler. As SammyS calculated, f(.5)= -0.9131< and f(1)= 1> 0 (f(x)= x+ ln(x)) so there must be a 0 between them. The simplest thing to do is to take half way between: 0.75. Now f(0.75)= 0.75+ ln(0.75)= 0.46231> 0. So there must be a root between 0.5 and 0.75. Half way between is 0.625. Continue until you have an accurate enough result.