Solving a Rational Exponential Equation

[@XJWill1]

For a more challenging problem, consider the case where the right-hand side of the equation is an arbitrary real constant, c: (1 + 1/x)^(x+1) = c This can be separated into two cases: c < e and c > e (c = e does not have a finite real solution) I will not show the steps, but the solution is: x = -1 / ( ln(c) / W( - ln(c) / c ) + 1 ) where W(u) is the product-log function, aka Lambert-W, and is: Wm(u), the lower branch of the product-log, for c < e Wp(u), the principal branch of the product-log, for c > e

[@SyberMath]

Np

[@XJWill1]

@@SyberMath He loves me, he loves me not. Love given and love taken away. So sad.

[@GourangaPL]

Story time: when i was at the university i missed my class where we were told about this e limit, so on the exam instead of just using this fact i covered 2 sheets of paper to proove it, then after the exam i talked to my classmates asking if they also found it out it was e and they were like "well yes it's equal to e, everyone knows that"... #emotional_damage :D

[@SyberMath]

😁

[@imonkalyanbarua]

Your solutions are so wholesome! Not only do you explain it analytically but also geometrically (or graphically, whichever you find comfortable) which helps us appreciate the depth of the problem and understand it in a more nuanced manner. Truly, you are doing an amazing work! God bless you.. 😇🙏

[@SyberMath]

Thank you for the kind words! 🥰

[@imonkalyanbarua]

@@SyberMath 🤗❤️

[@jesusrh3665]

If you look to the general case (x+1/x)^x+1=(1+1/k)^k,=> (x+1/x)^x+1=(1-1/k+1)^-k , just by comparing both sides you get the formula for the general case x=-(k+1).

[@cicik57]

limit are no use, you should calcualte derivative to show that the function always descends and therefore -7 is the only solution.

[@juanmolinas]

Wow Syber!, that´s really cool use to for (a/b)=(b/a)^-1...I had not thought of it...

[@SyberMath]

😊

[@pk2712]

The horizontal asymptote is y = e . The left branch approaches it from below ; but , never crosses it . The right branch approaches it from above but never crosses it , This is kind of what you said with the limits . y =(1+1/x)^(x+1) crosses y = (7/6)^7 at x = --- 7 because (7/6)^7 is less than e .

[@SyberMath]

Nice!

[@or6238]

Congratulations on 100k!!!

[@SyberMath]

Thank you so much 😀🥰💖

[@geraldtros835]

🙂 Thanks! Yes, I did find it interesting. And entertaining, as pretty well always!

[@SyberMath]

Np. Thank you!

[@urielozer]

Very smart solution

[@SyberMath]

Thanks!

[@markobavdek9450]

5:01 "... is getting smaller and smaller... But actually not...Anyways." Blablabla, philosophy...

[@pranavamali05]

Thnku

[@elmurazbsirov7617]

Təşəkkürlər.

[@SyberMath]

Rica ederim!

[@Jha-s-kitchen]

Nice algebraic manipulation of (7/6)^6 😂 I remember 3 digits of e, numerically 2.718 and 3.14 of pi

[@SyberMath]

You are good!

[@Jha-s-kitchen]

@SyberMath, now upto 5 digits, 3.14159 after watching this: kzbin.info/www/bejne/r53Ff5lvg655i8k 😂😂🤣

[@Muslim_011]

There is only one solution x=-7 The e is not a solution. The f(x) aprouch the e line but not crossing it at all

[@yakupbuyankara5903]

X=-7.

[@barakathaider6333]

👍🌹

[@SyberMath]

💖