Quandaries and Queries


I have the following equation:

y = ln(x) + x

How do I solve for x?
Thanks in advance

Question for secondary



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

y = f(x) = ln(x) + x

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

in the expression y = log(x) + x change x to y and y to x and then solve for y.

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 X-axis becomes the Y-axis and the Y-axis becomes the X-axis.

The final algebraic instruction "solve for y" means, geometrically, "put the graph in standard position with the X-axis horizontal with positive to the right, and the Y-axis 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,


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