Quandaries
and Queries 

Hello,
How do I solve for x? 

Hi Alain, This is a nice question. The simple "flippant" answer is, you don't, but there is a lot more you can say. I want to name this function using functional notation so I am going to use
The challenge is to solve for x, that is to express x as a function of y. We usually think of x as the independent (input) variable of a function and y as the dependent (output) variable so I am going to change the instructions by saying
So I have x = log(y) + y and the instructions are to solve for y. If I could do it this would give an expression in x which I am going to call g(x). I can't do it algebraically but I can do it geometrically. First plot the function y = log(x) + x. The domain of the function is all positive real numbers so I plotted it for 0 < x ≤5.
Next change x to y and y to x.
Notice that the variable names get changed in the function expression but also the Xaxis becomes the Yaxis and the Yaxis becomes the Xaxis. The final algebraic instruction "solve for y" means, geometrically, "put the graph in standard position with the Xaxis horizontal with positive to the right, and the Yaxis vertical with positive upwards". Rotating the graph and flipping it I get
The point I am trying to make here is that y = g(x) is a perfectly good function, we just can't write it algebraically in a convenient firm. To be more precise g(x) can not be written in terms of polynomials, roots, logarithms, exponential functions or trigonometric functions. I hope this helps, 

