An Interesting Non-standard Equation
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Thanks for a all the effort, as per usual.
This certainly looked like fun, so I gave it a shot before watching the video. (As I usually do, I just don't comment that often.) I used a syntax that can be input into Wolfram Alpha easily here, just chop off the part with the ";" at the end if anyone cares to check parts.
x^45 = 243^(x^2); prime decomposition
x^(3^2*5) = (3^5)^(x^2); redistribute power
x^(3^2*5) = 3^(x^2*5); ()^(1/5)
Note: taking the quintic root does get rid of 4 complex intermediate branches for this step of solving the eqation in this kind of domain. The root is of an odd value though, so there isn't a second real root (that would be the unary negated version of the first output).
x^(3^2) = 3^(x^2)
By direct inspection, x = 3 .
Is there a second real solution? Try to arrange the equation for the Lambert W function.
x^(3^2) = 3^(x^2); ln()
3^2*ln(x) = x^2*ln(3); ()*x^-2*3^-2
x^-2*ln(x) = 3^-2*ln(3); use e^ln(x^-2) = x^-2
e^(-2*ln(x))*ln(x) = 3^-2*ln(3); -2*()
-2*ln(x)*e^(-2*ln(x)) = -2*3^-2*ln(3); use e^ln(3^-2) = 3^-2
-2*ln(x)*e^(-2*ln(x)) = -2*ln(3)*e^(-2*ln(3)); apply Lambert W function
-2*ln(x) = ProductLog(-2*ln(3)*e^(-2*ln(3))); -1/2*()
ln(x) = -1/2*ProductLog(-2*ln(3)*e^(-2*ln(3))); e^()
x = e^(-1/2*ProductLog(-2*ln(3)*e^(-2*ln(3))))
Again, by direct inspection, x = 3 for one possible answer.
Wolfram Alpha tells that "e^(-1/2*ProductLog(0, -2*ln(3)*e^(-2*ln(3))))" with an output value of near 1.188 is the the primary solution.
The expression "e^(-1/2*ProductLog(-1, -2*ln(3)*e^(-2*ln(3))))" for the next branch equals 3, as expected.
All other expressions for the Lambert W function will yield (countable) infinitely many complex solutions. From the earlier discarded 4 branches (by taking the quintic root) all results will yield sets of just complex solutions, too.
Very nice!
How are numerical values calculated from the Lambert W function?
See the Wikipedia page, it has an infinitely large series expansion. There are easier to approximate functions used in calculators for numerical outputs though.
You should make a UA-cam channel called LambertWFunction.
Good idea!
x = 3
x=e^(-0,5W(-ln9/9))=1,188...oltre al banale 3
Nice!
Thanks!
x^3^15 x^3^3^5 x1^3^1 x^3^1 (x ➖ 3x+1) 3^5x^2 3^2^3x^2 1^1^3x^2 3x^2 (x ➖ 3x+2).
Pretty gross, answer better be 3 or we might be in trouble.
O_o
@nightstar8444 j/k, there a W function for that