Exponential Function. Polynomial Question. Algebra Problem || Lambert w rule. Math Olympiad Exam.

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[tombufford]

A quick look at this and 3*3*3 = 27 . 27 + 3 = 30 hence x = 3. This helps check the answer learning and using the Lambert w rule.

[ssalmero]

x=3 is an obvious solution. Then you need to prove that this is the only one solution. As 3^x is an increasing function and 30-x is a decreasing function, this demonstrates x=3 is unique solution.

[johnnath]

Re-write as f(x) = g(x), where f(x) = 3^x & g(x) = 30 - x; f(x) is a strictly increasing function & g(x) is a strictly decreasing function, so f(x) & (g(x) can intersect in at most one point. But by inspection x = 3 is a point of intersection (since 3^3 = 27 = 30 - 3), so x= 3 is the unique solution.

[wes]

I just look at this equation and see 3^3 + 3 = 30. If I wrote that down and said x = 3, would the teacher give me credit?

If the problem didn't have such a simple answer I would employ an iterative Newton-Raphson solution method.
let f(x) = a^x + x – b where a and b are specified real, positive numbers allowing a real value for x satisfying f(x) = 0. The N-R solution method is x ← x – f(x)/f'(x) where f'(x) = a^x ln(a) + 1 and ← means to assign the value on the right to the variable on the left. The N-R solution method requires a rough starting estimate for x that I would get from a rough plot of f(x).

[nasrullahhusnan]

Take modulo 2: 1^x+mod(x, 2)=0
mod(x, 2)=-1 --> mod(x, 2)=1
x=2k+1 where k=any integer
Take modulo 3: mod(x, 3)=0
x=3k where k=any integer
3k=2k+1 --> k=1 and x=3

[kimba]

Wait, I'm lost.
"When you apply that calculator you will find that x equals to 3". I could have done THAT from the start. And of course, WolframAlfa is readily available in International Math Olympiad Exams.
3^x is strictly increasing, and 30-x strictly decreasing, therefor there is one solution. x=3 is obvious by inspection

[olusolaomolayo]

How do you use calculator to find Lambert wn function of a term like the one that gave 27 according to you

[에스피-ht]

let f(x)=3^x g(x)=30-x
The two graphs meet on only one point of (3, 27)
Answer x=3

[souzasilva]

How to use Lambert's formula to determine all roots of an equation, if any?

[zakzakzak]

Зачем такие сложности... Очевидно, что x>1. При этом x функция монотонно возрастает, следовательно существует только 1 корень, который очевидно равен 3. Задача решается в уме за три (3) секунды!

[walterwen]

Math Olympiad Exam: 3^x = 30 – x; x = ?
Simple solution:
3^x + x = 30 = 3(10) = 3(9 + 1) = 3^3 + 3; x = 3