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In a recent MathCounts Contest, the following question was asked

If $x$ satisfies $4^x + 5^x= 6^x$ then find the greatest integer not greater than $x$

I tried to take logarithms of both sides, but then couldn't figure what to do with the $\log (4^x +5^x)$ part. Next, I tried to do some algebraic maneuvers to match the bases but failed in doing so.

How do I solve this problem and generally speaking, how do you tackle equations when the variable is in the exponent part?

(The time limit for solving the problem was 1 min and one guy solved it in under 10 seconds, which made me really curious, did he know the question from beforehand, or is there some kind of strategy?)

Anurag Saha
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    Just try small integer values for $x$. That tells you the answer very quickly. Remember, you are not asked to solve for $x$, just to get integer bounds on it. – lulu Apr 12 '19 at 20:15
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    $4^2+5^2>6^2$ and $4^3+5^3<6^3.$ Proving there is only one such $x$ is another problem. – Thomas Andrews Apr 12 '19 at 20:15

1 Answers1

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Hint: After dividing both side by $5^x$, we have $$\big(\frac45\big)^x+1=\big(\frac65\big)^x$$

Now, the left side is decreasing(power of a proper fraction) and right side is increasing(as, $6/5$ is greater than $1$) for $x>1$. So, maximum one root possible.

tarit goswami
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