Indian Olympiads Entrance | A Nice Olympiads Trick | How to Solve for x?
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- Опубліковано 5 вер 2024
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Great!
Another approach:
5ˣ = x⁶²⁵
xln5 = 625lnx
x/lnx = 625/ln5 = 5⁴/ln5 = 5⁵/(5ln5)= 5⁵/ln(5⁵) So x/lnx = 5⁵/ln(5⁵) = 3125/ln(3125)
which means that x = 3125
The horizontal line y = 3125/ln(3125) intersects with y = x/lnx at two values of x.
x = 3125 and
x ≈ 1.00258509314089
Note that
5ˣ ≈ 5.02084607
x⁶²⁵ ≈ 5.02084607
So there's a second solution.
This implies that 3125 is a solution. But are there any others ? There is definitely one between 1 and 625, since 5^1>1^625 and 5^625
Nice insight! The second solution is approximately
x ≈ 1.00258509314089
Note that
5ˣ ≈ 5.02084607
x⁶²⁵ ≈ 5.02084607
@TheOldeCrowe Now we have two solutions. Are there any more ? If so find them all.If not, prove that. Only then problem will be fully sllved
@marklevin3236 I think there should be a way to prove there's only two solutions. Based on the behavior of the exponential and odd power functions (both concave up and monotonically increasing for x > 0), their function plots should not cross a third time.