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I found a question in a practice GRE Mathematics test, but I wasn't sure if I was using the right method to solve it. The question is to find the number of intersections in the x-y plane of the functions 2^{x}and x^{12}. So I got;

[tex]\frac{\ln{2}}{12} = \frac{\ln{x}}{x}[/tex]

I don't think a calculator is allowed for this paper, but I was wondering if this was the right method. I was going to rewrite ln(x) as a Taylor series approximation, divide the whole thing by x and then see where that got me. Is this the wrong way to do it?

Other than that I'm not sure how to do it. Given that it is in an exam it can't require a computer program to solve it either... I would appreciate some help on this. Plus, I don't know if the graph of 2^x may 'overtake' the graph of x^12, or not... so, not sure what to do. Help would be appreciated.

The possible answers are;

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

I'm pretty sure A is wrong, but I don't know about the rest. If I had to guess I would say at least 2.

You also only get roughly 2.5 minutes per question in this test (66 questions in 170 minutes) so I'm not sure if this method is even practical... is there something obvious I'm missing? I know the general shapes of both graphs, so if one were to overtake another, the derivative of 2^x would equal the derivative of x^12 at some point and then it would be less than the derivative of x^12 after that, right? But even if that were true that would just give me one root.

EDIT: Also, this is not a homework question, just bored in my summer holidays.

I thought about the graphs of Kx and ln(x) and I'm more convinced there is 1 intersection pretty early on (at x = 0, Kx is at origin, ln(x) is rising very fast). I think the answer is 2 or 3.

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# 2^x = x^12

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