A Nice Math Olympiad Exponential Equation 3^x = X^9

A Nice Exponential Equation 3^x = X^9 How to Solve Math Olympiad Question 3^x = X^9 Exponential Equation? What is the value of x? Find the value of x? Solving Math Olympiad Question A Nice Exponential Equation 3^x = X^9

In this video, we'll show you How to Solve Math Olympiad Question A Nice Exponential Equation 3^x = X^9 in a clear and concise way. Whether you're a student learning the basics or a professional looking to refresh your knowledge, this video is for you. By the end of this video, you'll have a solid understanding of how to solve math olympiad exponential equations and be able to apply these skills to a variety of problems. So whether you're studying for an exam or just looking to improve your math skills, be sure to watch this video and take your math knowledge to the next level!

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#Maths

[sergeyzelensky]

Nice, but...when we are talking about solving equation, it is necessary to find all of them or indicate that it's required find some of x. From graphs of functions easy to see that the equation has two solutions. One is : 1.15<x<1.16. Please keep attention to this.

[Altair]

Nice trick, but there's a flaw in the end when you deduce from x^(1/x) = 27^(1/27) that x=27. There's no warranty that it's the only solution, you'd have to prove that the function f(x) = x^(1/x) is injective for that. And it's not the case. It's the same mistake as when you have x² = 2², and you deduce from it that x = 2. You missed that x = -2 was also a solution.

[davidbrisbane]

The two real solutions using the Lambert W function are x = 27 and x = 1.150825 (6 decimal places). There are of course and infinite number of complex solutions as well.

[leodouskyron]

The funny thing is the note at the bottom saying you should know this … pride comes before the fall when you are dealing with math geeks and there can be multiple solutions.

[instinx]

just cube each side and you get 3^(3x) = x^27. Then, (3^3)^x = x^27. Finally, 27^x =x^27 and then it's really easy to see x=27 is a solution. However, 27 isn't the only solution as another commenter already pointed out.

[isaacchen]

It's easy to see x > 0. Now, notice the given condition is equivalent to x ln(3) = 9 ln(x) or ln(x) / x = ln(3) / 9 = ln(27) / 27. Because the first derivative of ln(x) / x is (1 - ln(x)) / x^2, we know ln(x) / x is increasing on (0, e) and decreasing on (e, infty). Thus, observing that x = 27 > e is a solution implies that there is one other solution in (0, e).

[binishulman]

What I'm not happy about is that your method of solving required that you knew 27 would be the solution in the first place.

[zihaoooi]

doing this before the video
x^-9 * 3^x = 1
x^-9 * e^ln 3^x = 1
x^-9 * e^x ln 3 = 1
(x^-9)^-1/9 * (e^x ln 3)^-1/9 = 1^-1/9
x e^-(x ln 3/9) = 1 | you might have noticed i only used one of the roots. The W function actually handles the rest of these roots. im using the principal for now.
-(x ln 3/9) e^-(x ln 3/9) = -ln 3/9
Lambert W Function:
W(-(x ln 3/9) e^-(x ln 3/9)) = W(-(ln 3)/9)
-(x ln 3/9) = W(-(ln 3)/9) | W(xe^x) = x
x = (-9W(-(ln 3)/9))/ln 3

[rushivyas]

@1:32 you already know the answer that's why you multiplied & divide by 3.

[Mrcometo]

You are missing solutions. If you try x=1 and x=2 you can see that there is another solution between these values of x.

[eulerthegreatestofall]

Using Lambert W function, I got the following solution: x=exp^(-W(-Ln[3]/9))=> x=1.15082 which is not the only possible solution. But, it does satisfy the equation.

[ajejebrazov]

Sorry, but you cannot just use as condition that base and exponents must be equal on both sides. It is not the only solution of the problem

[joshdeconcentrated]

I would avoid posting incorrect math. next time don't assume injectivity.

[digitalbroadway]

正确的解题思路和方法是对等号两边同时取Log, 将复杂的指数形式变成对数形式，经运算后即可得到解。

[divyeshjoshi]

I am starting to feel that random things appeared just to prove LHS = RHS

[helwasyaharinramadhan]

Let x = 3^t, so 3^t = 9t
Let t = 3^s, so 3^s = 2 + s → s = 1
So t = 3¹ = 3, then x = 3³ = 27

[recramorcenlemniscate]

Why are you able to multiply both the radicant & index by 3? Just to get the equation to work?

[Danin]

You have a logical fallacy when you say “you should know this”. Most of the regular folks clicking on this video don’t know ‘this’. 😊

[philippegilles]

Hi, thanks for the clever trick, but ... As many people already said, it's not the only solution : ~1.1508 is also a valid solution.

Also, at 1:34 how are we supposed to know when to do that? It's very interesting but I'm never gonna know when and how to use that 😢.

PS : thank you anyway

[Deto]

Is this really a trick to solve the problem or just something that's straightforward once you already know the solution? It's like showing someone how to solve to a maze as - "just turn here then here then here!"