A Very Nice Exponential Equation | Math Olympiads

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x^x=2^{3x+192}
#ChallengingMathProblems #ExponentialEquations
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[kicorse]

If you substitute 2^a for x, you quickly get (a-3)2^a = 192, from which a=6 and x=64 drops out. You still need to look at the function to show this is a unique real solution of course.

[mcwulf]

Factor out the 3 to get
X^x = 8^x . 8 ^64
Divide by 8^x
(X/8)^x = 8^64
Take 8th root
(x/8)^(x/8) = 8 ^8
So (x/8) = 8
x = 64

[paulnokleberg]

I also used the lambert w function and got 8 times e to power w of ln(2^24). It makes sense that if you take w of ln of (a^a) and then raise e to that power, you get a. I didn't simplify 2^24 to 8^8 at first, otherwise it would have been obvious that x/8 equals 8 and x equals 64. The lambert w expression is correct, just not simplified.

[cicik]

assume that answer is integer, it will have a form x^x = 2^(n 2^n) so
n 2^n = 3*2^n + 192.
(n-3) * 2^n = 192
the value of n is bounded from above so 2^n < 192, and we must check only a few values and get n = 6, x = 2^6.
Lets take that x can be negative and even, so x^x > 0 .
x^x = (-2^n)^(- 2^n)
-n^2^n = - 3*2^n + 192
(3-n)2^n = 192
maybe there is other not integer solution here?

[MortezaSabzian-dbsl]

Well, to solve this problem, we can use the Lambert function
X^x=2^(3x+192)
First, we try to make both sides of the equation symmetric and make it so that after using the Lambert function, there is only one variable in the equation.
(X/8)^(x/8)=8^8
We take the logarithm of both sides
Ln(x/8)(x/8)=ln(8)(8)

We put both sides in the Lambert function

We should know something about the right side of the equation
If the Lambert function input is between -1/e<a<0 Let the output of the function be two numbers and we have two answers, but if the input of the function is greater than zero, we have one answer
So 0<8ln(8)
So we have an answer
ln(X/8)=ln(8)
X=64

[michaeltrungold]

I got x= 8e^(W(24ln(2))), where W is Lambert w. The principal branch gives 64, but other branches give alternate, complex solutions

[jonathanward]

A few days ago Sybermath solved x^2 + x^2 / (x + 1}^2 = 3. The use of substitution was dazzling, so much so the following night I dreamed the video over and over. Women are still the most beautiful thing in the world but now I dream obout math substitution.

[benismann]

7:18

that's kinda not hard to guess. Idk maybe im weird, but the minute i saw it i thought "3x+192 should be a square of some number. X should also be 2^N where N is whatever idk yet, but probably more than 4" so the first thing i thought of was 32, but 288 is 144*2 so it's not a square. Then i thought of 64 and boom

[goldfing]

x^x = 2^(3x + 192) = 2^(3*(x + 64)) = (2^3)^(x + 64) = 8^(x + 64) = 8^x * 8^64
Divide by 8^x:
x^x / 8^x = 8^64
(x/8)^x = 8^64
By comparison of the exponents, I assume x = 64 could be a solution. Indeed,
(64/8)^64 = 8^64
Hmmm, don't know if there are more solutions.

[knutthompson]

Good ol’ x^x again. There are other complex solutions

[lucas_]

I thought it was going to be easy but when I realized I couldn't even isolate the x xd

[Chris_]

Nice video, but which app do you use to write?