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?)
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.