(homework) exponential equation problem

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vhunter
Posts: 1
Joined: Tue Feb 18, 2014 10:33 am UTC

(homework) exponential equation problem

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.

LaserGuy
Posts: 4581
Joined: Thu Jan 15, 2009 5:33 pm UTC

Re: (homework) exponential equation problem

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.

alessandro95
Posts: 109
Joined: Wed Apr 24, 2013 1:33 am UTC

Re: (homework) exponential equation problem

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)
The primary reason Bourbaki stopped writing books was the realization that Lang was one single person.

Forest Goose
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Joined: Sat May 18, 2013 9:27 am UTC

Re: (homework) exponential equation problem

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.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

Bloopy
Posts: 215
Joined: Wed May 04, 2011 9:16 am UTC
Location: New Zealand

Re: (homework) exponential equation problem

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.