(homework) exponential equation problem

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(homework) exponential equation problem

Postby vhunter » Tue Feb 18, 2014 10:55 am UTC

I'm helping my brother with his homework, but this one got me stumped. The answer (x=1) has been provided, but I can't figure out how to get there. I've tried taking the log of both sides, but that didn't seem to simplify the issue. Could you help?

The question is: Find the value(s) of x for which 2^x+5^x=7^x.

I'm not sure whether standards are the same internationally, but this question is intended for students in their last year of high school.

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Re: (homework) exponential equation problem

Postby LaserGuy » Tue Feb 18, 2014 7:06 pm UTC

Well, you can get the answer x = 1 by inspection. There isn't really a nice way to solve this algebraically, so proving that is the only solution is a bit trickier.

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Re: (homework) exponential equation problem

Postby alessandro95 » Tue Feb 18, 2014 7:40 pm UTC

Thanks to the Fermat-Wiles theorem (Fermat's last theorem) you only need to try x=1 and x=2 to decide that there's only one integer solution, but I don't think this is how he was supposed to solve the question (I'm in the last year of high school too, but I have no idea on how to solve a similar equation in the general case)
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Re: (homework) exponential equation problem

Postby Forest Goose » Tue Feb 18, 2014 8:20 pm UTC

Divide by 7^x, you get an equation of form p^x + (1-p)^x = 1. It's easy to see that x =1 is a solution; and since the left side is decreasing for positive x, it must be the only solution.

You can use calculus to see that it's decreasing by taking the derivative and observing it is negative. But, there is a far simpler way: a^x is decreasing for 0 < a < 1, since p and 1-p are both in this interval, the function is strictly decreasing for x >= 0, so the solution is unique.

*to understand why strictly decreasing => uniqueness: if f(x) = a and f is decreasing, then y < x implies f(y) > a and x<y implies f(y) < a, thus f(y) cannot equal a for y !=x.
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Re: (homework) exponential equation problem

Postby Bloopy » Wed Feb 19, 2014 2:27 am UTC

When you take the log of both sides, what you do get is a picture of their relative exponential growth rates.

You could observe the growth factors. The growth factor of 2x + 5x is limited to between 2 and 5. So 7x always grows faster (or when x is decreasing, 7x always decays faster). Therefore, they must meet at only one point.

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