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Equivalent (and imho nicer) solution is 2*sqrt(W(1/4)). You can get there from your solution with some W identities, but I found the algebra simpler when I started by exp-ing both sides of ln x = - x^2/8. Briefly: ln x = -x^2/8 exp-ing: x = exp(-x^2 / 8) div by x: 1 = x^-1 exp(-x^2 / 8) pow to -2 (*): 1 = x^2 exp(x^2 / 4) div by 4 and W: W(1/4) = x^2 / 4 Solve for x: x = +- sqrt(W(1/4)). An extra root was artificially introduced in the (*) step above so we can discard the - sign if we want to stay in R.
Interesting, it was not intuitive to me that the first equation is actually y = x^(4-(x^(4-(x^4-(..., rather than x^(4-x)^(4-x)^(4-x)^.... If written the second way, then you cannot do the y = x^y simplification. It was not clear that the power tower was balanced upon just the x and not the 4-x.
This says that the limit of (any expression raised to a power) is equal to (the limit of the expression) raised to the same power. Is this always true?
There is something within me that might be illusion as it is often case with young delighted people, but if I would be fortunate to achieve some of my ideals, it would be on the behalf of the whole of humanity. If those hopes would become fulfilled, the most exciting thought would be that it is a deed of a Serb." Nikola Tesla, visiting Belgrade in 1892.
W is just an approximation like log. Nothing special about it. Now it is better if you use a computer numerically solving such equations, you do it at the end anyway by calculating W.
You could also simplify the answer to -2ln(2) / W(-2ln(2)) and/or change -2ln(2) to -2ln(±2), which can give you both the solution in the video and -2, which is also correct.
hi friends. hi teacher We know that W(xlnx) = lnx And how about *W[(lnx)/x]* ? Is there some "formula" in this case? For example We know W(17ln17) = ln17 But to find/calculate W[(ln17)/17], can we find an exact value, a perfect value, without using approximations, without using things like Wolfram Alpha?
Yes, it was interesting. Unfortunately, in university, I never had this problem. Backt then I loved math but not physics. I still love math and find physics much more interesting
