Solving a 'Harvard' University entrance exam

How can we solve this?

Higher Mathematics
Prime Newtons
Lambert W function should be new elementary
Wiki history
Lambert W
SE solving linear = exponential
WolframAlpha

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[brendanward]

It's called the W function because in the end you need to use Wolfram-Alpha to solve the equation.

[LiteBulb]

I should've tried this technique on tests where I couldn't figure things out. "The answer is B(5), where B is a function I'm defining right now that will solve this problem."

[luisfilipe]

I’ll never not be amazed by mathematicians ability to just make stuff up and call it the day

[chuckw]

So it still can't be solved by hand and needs a computer/calculator and I still don't know what a Lambert function is. I'll call it a day.

[michaelz]

Learning about the “Goal Seek” feature in Excel alone was worth the cost of admission. Thanks!

[cguy]

I think people are missing the fact that the Lambert W function is not just some arbitrary inverse, otherwise Presh could have just said P(2^x+x) = 5 and stopped there. The Lambert W function has been extensively researched, has a lot of properties, and identities, and is quite useful. This is why Presh went to the trouble to reformulate the problem into the product-log form.

[martinhertsius]

What's the point of all this when there is no explanation of what the W function does??

[krabkrabkrab]

In my head, I tried x=5/3 and realized it's a bit low. SO I went for 1.7. Then Newton's method: x_new= x- (x immediately gives 1.7156 (on a calculator that doesn't have a Lambert function).

[verkuilb]

Let me get this straight—you follow up a video about whether 3x5 is the same as 5x3…with this???
🤯

[asparkdeity]

And to those complaining, we got a near identical question in our Cambridge maths entrance exam, the very paper I sat had a question with the lambert-W function. Don’t believe me, look up STEP II 2021 Q4. Not something I had ever learnt in school or heard of at the time, but given its introduction I was still able to do the question.

It’s not about solving the question for an exact answer using a calculator, but it’s about understanding and applying new techniques to gain an analytic closed form solution to an unseen problem. It actually tests your true mathematical ability.

[chrisarmstrong]

The Lambert W function was never mentioned in my High School or University maths subjects (in the 1970's !). Thanks for the info.

[christianbohning]

It is worth to mention that the Lambert-W function isn't exactly one function. To invert x * e^x in the real domain one needs two different branches of the Lambert-W functions, otherwise there would be two function values for x between -1/e and 0. Meaning that for x between -1/e and 0 only one of the two function branches might give you the desired solution, and in that case it's pretty tricky to know which one. Also, x < -1/e doesn't yield any real solution.

[JonSebastianF]

*_U 2 to the Power of U_*
...sounds like a power ballad by Prince💜

[wernerviehhauser]

ok, but could we just NOT do a Lambert W Function for a week or so? The videos on that topic are getting out of hand...

[danmerget]

I solved the problem in a slightly different way, and got x = log2( W(32 * ln(2)) / ln(2) ). When I plugged it into a calculator, I got the same result as Presh: 1.71562.

I was a bit freaked out as to how two different-looking answers could give the same result without any obvious conversion between them, but then I noticed that both answers contain W(32 * ln(2)) / ln(2). If we call that quantity Y, then Presh's answer was x = 5 - Y, and mine was x = log2(Y). The only way these two answers could be the same is if Y = 5 - x = 2^x, which would imply that 2^x + x = 5, and oohhhh I get it now.

[mr]

@Blackpenredpen does a lot of videos (think a whole playlist's worth) re: Lambert W function and explains it rather well... Bonus - he also uses "fish" to explain it! 😂

[mikeymcchoas]

I lost my shoes once. Couldn't find them anywhere. Few weeks later, I'd forgotten that I lost them and went and got them.

[ManjeetRani-vn]

bro humiliated me (an indian 9th grader) in every single way by saying, "i wasn't able to go to harvard, that's why i went to stanford 0:23 ". btw: thanks for uploading such glorious content, your daily uploads makes my day, everyday.

[Smallpriest]

For everyone complaining, consider ln(5) (natural log)

If the answer was ln(5), would you say that it's an exact solution?

If so, why would W(5) (lambert W) not also be an exact solution?

[crimsoncanvas]

High school maths to solve is assume f(x) = 2^x+ x-5 and use Newton raphson method.

xn1= xn0- f(xn0) /f'(xn0)