This question has come up this morning in the office and we’re stumped for an answer. Can this equation be solved algebraically?
x[sup]x[/sup]=27
One can see intuitively that x=3 and it’s also possible to solve graphically. But algebraically?
If by algebraically you mean have one x on one side and a number on the other, I don’t think so. You could take the log:
xlog(x) = log(27)
or x= log(27)/logx
It might look nicer, but I don’t think it helps a lot
**Splanky ** - yes, by *algebraically * I meant manipulating the equation to end up with x as the subject on one side. We tried taking logs, but as you say, it doesn’t really help very much.
There may be some fancy trick, but for the most part, I think you’d have to keep computing x[sup]x[/sup] until you hit your target or pass it.
Take the xth root of both sides.
Now x = xth root of 27
Solve by using a spread sheet with reasonable values of x and home in on the answer. There may be a more rigorous way to be furnished by an advance math major student.
As a math grad, albeit a crusty and rusty one, I believe there is some substitution you can use to solve it, but since I haven’t used it in 15 years or so, I forgot it.
There is indeed no simple closed form. The equation that some have shown above after taking logs:
t log t = n
appears quite frequently in the analysis of algorithms in Computer Science. E.g., in this case t is time and n is input size. And expressing t in terms of n is what we really want. In Computer Science we are usually happy with “Big Oh” order notation and it turns out that the following approximation is plenty close enough for our purposes:
t ~ n / log n
Also note that if you are trying to solve this numerically, even lame old techniques such as Newton-Raphson iteration are far superior to what has been suggested so far.
It sounds like what you’re looking for is an “elementary inverse” for the function f(x)=x[sup]x[/sup]: i.e., a closed-form expression in terms of elementary functions (trigonometric and exponential functions and their inverses) for f[sup]-1/sup. (Then you could just take x = f[sup]-1/sup.) As ftg says, there is no such elementary inverse. The Lambert W function is defined as W = f[sup]-1[/sup], so if that’s implemented on your platform of choice, then you can use that.
No, there’s no close-form solution. But it can be solved iteratively. Set up the function y = x[sup]x[/sup]-27 and find the root iteratively, such as with Newton’s method. To find the derivative of y, you can use implicit differentiation.
I just break out my trusty K&E LogLog Duplex Decitrig… This is one of the few things still easier to do with the slipstick than a calculator (assuming you have the right sliderule and don’t require great accuracy). If you can get your hands on one of these, it can help you to understand a great deal about logarithms. It did for me.
As a nitpick to add to the negative results, to some solething algebraically means to do so in polynomials. Taking logs automatically removes you from the algebraic realm.
“solve something”. Brain hurts over these damned polls.
The academic convention is to blind us with algebra and logic, but guess what? They hide from us the fact that some problems are solved by iterations. One well-known method for this is Newton’s method. So, iterations have been a tool of the trade for a long time, but our “high-and-mighty” academics fail to teach us to think outside the box and there’s more than one way to skin a cat. We need to teach people such thinking because it is just as valuable when tackling problems.
- Jinx
Newton Raphson and other iterative methods are taught at AS level in the UK. They’re hardly a secret.
Christ, take a pill! No mathematician would say that there’s only one way to reach a correct answer (Serge Lang would say there’s only one correct way to reach a correct answer and many incorrect ways). We very much teach solutions by approximation and things like Newton’s method. In fact, I’m willing to bet the OP knows about such solutions, especially given the comment about solving the problem “graphically”, and is asking just if there is a closed-form solution.
There is no vast math-wing conspiracy.
Thanks for the comments everyone. It looks like iterative and graphical solutions are the only possibilities.
Sorry if I misused the term “algebraic”, Mathochist. It’s twenty years since I’ve studied any maths. 
As I said, it’s a nitpick. Most people wouldn’t know it, but we are fighting ignorance here.
despite major setbacks…
Oh, there are a number of methods. Newton-Raphson (probably the best method) is iterative, and there are surely other iterative methods, but I’m not sure I’d describe, say, the method of bisection that way. There might also be some sort of series expansion for the Lambert W function, which you could use, and which also would not be iterative.
Did you know that pretty much every math department across the country offers courses in numerical analysis?
x[sup]x[/sup] is an interesting function. It has a minimum on x>0. What is the value of the minimum and where does it occur?