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I don't believe it has a solution. I shall provide some graphical and analytical support for this claim.
To start, I began by attempting to simplify the expression in order to see if a solution could be seen that way. After a few steps, I arrived at the following:
I did not see a way that this could be solved, so I defined this expression as a function f:
Now the issue at hand is to see if f(x) is ever zero. Since algebraic techniques did not help in solving the problem, I graphed this function to see if it had a root. In fact, the lowest it seemed to go was approximately 600.78397, when x = 1.746827. Now the remaining problem is this: Is 600.78397 the only minimum value on the graph of our function f(x)?
To answer this, I will use calculus. I'm not sure if you're comfortable or not with this, but I will go ahead just to show that the problem has no solution. Now take the first derivative of f(x):
We know that the graphically observed point 1.746827 is a minimum, because df/dx | x = 1.746827 is zero. We have already graphically confirmed this, however. The real question is whether or not there are more minimums on f(x) which would possibly be low enough to allow a value of 0 for f(x). To investigate this we examine the second derivative:
Note that the second derivative is always positive, since the square of the natural logarithm is always positive, and a positive value taken to an exponent always results in a positive answer. Since the second derivative is always positive, then f(x) is concave up on (∞, ∞), which means its slope is always increasing. A result of this is that there will only be one minimum to f(x). But we have already found a minimum of f(x), as this can be the only minimum, there is no value for f(x) that is less that 600.78397, so it has no roots and thus the original equation posted has no solution.
The problem is that ∞ - ∞ can be anything. Consider the following examples, which are limits in the form ∞ - ∞:
Another example with a different result is as follows:
These two examples serve to support the following statements: ∞ - ∞ is not necessarily ∞ or 0. Since infinity is more of a concept and not an exact value, there are infinitely many infinities. Several different infinities were seen in the above examples. Because of this quality of ∞, ∞ - ∞ has infinitely many solutions.
Here's an interesting one:
111111111² = 12345678987654321.
Basically, when any number composed entirely of n ones is squared, the result will be 12...n...21. Well, at least when n < 10, once we get to 10 the middle gets screwy.
I was almost convinced that 0 x ∞ = 0 x 1/0 = 1, now I'd have to tell my mind it is ±1.
With "∞ = 1/0" reasoning, we could argue that 2/0 is also ∞, and so on. From this, 0 x ∞ could be any number. Letting n/0 = ∞ is just illegal to me. True, n/m gets increasingly closer to ∞ as m gets closer to 0(from the right), but when m actually is 0, we don't have any sensible number to assign to it. Assigning ∞ to n/0 doesn't make any sense unless n = 0, making it so 0 x ∞ = 0, but then we run into the classic indeterminate form 0/0, which can be any number. As mathsyperson said, dealing with 0 like this brings us into an area of mathematics which we cannot explain.