Solving an Interesting Exponential Equation

🤩 Hello everyone, I'm very excited to bring you a new channel (SyberMath Shorts).
Enjoy...and thank you for your support!!! 🧡🥰🎉🥳🧡

If you need to post a picture of your solution or idea:

#exponentialequations #algebra
via @YouTube @Apple @Desmos @NotabilityApp @googledocs @canva

PLAYLISTS 🎵 :

[mbmillermo]

I felt like you never gave us the answer. Isn't it x = n^(1/n) ? I think it was implied but never stated. You got the t=n, but then didn't go back to the x. I think you then focused on whether there was only one solution and forgot about x.

[Okkk]

Correct me if I am wrong, we can replace n in the exponent with x^x^n recursively to obtain n=x^x^x^x^x… which can be expressed as n=x^n or x=n^(1/n).

[XJWill]

Incidentally, I think a more interesting question would have been: for x^(x^n) = n where n is any positive real number, what is the maximum value of x possible?

[justabunga]

All you have to do is to raise both sides by n, which is x^(nx^n)=n^n. The base and the exponent must match on the right hand side, which is (x^n)^(x^n)=n^n. We can tell that x^n=n, so x=n^(1/n) or nth root of n.

[oenrn]

The answer is always nth rooth of n, regardless of the number of x's.

x = n^(1/n)
x^n = n
x^(x^n) = x^(n) = n
x^(x^x^n) = x^(n) = n
...
x^(x^x^x^x^...^x^n) = x^(n) = n

(redundant brackets used to make it clear where the substitution happens)

[XJWill]

No, that does not make sense. A couple things in the video do not make sense.
FIrst of all, for real n between 0 and 1 except for n = exp(-1), there are two real values for x that satisfy the equation. They are x = n^(1/n) and x = (n*ln(n) / wm(n*ln(n)))^(1/n) where wm() is the negative real branch of the product-log function (aka Lambert's W). For real n > 1, there is only 1 solution, x = n^(1/n)
Also, the explanation for why you don't use the second derivative makes no sense. When looking for extrema, you do not set the second derivative equal to zero as you seemed to suggest. What you do is check whether the second derivative is positive (concave up), negative (concave down), or zero (saddle point) at any x value where the first derivative is zero. That tells you whether the point in question is a minimum, maximum, or neither.