A Homemade Exponential Equation | SyberMath

[@dmtri1974]

Really nice method! Just one minor remark: in order to be sure that there are only 2 solutions (in Reals) we need to study the the function: f[x]= x^x * 4^(x+1/16) for x>0 , It is descendind in the interval (0, 1/4e) and ascendind in the interval [1/4e, Infty) and lim f as x-> 0 is 4^(1/16) >1 and lim f as x-> Infty is Infty >1 so there are at most 2 solutions which you found.

[@vbeckerlima]

What is this program do you use to make graphs??

[@SyberMath]

Desmos

[@vbeckerlima]

​@@SyberMaththx

[@SyberMath]

@@vbeckerlima np

[@elederiruzkin8835]

Wonderful!

[@popitripodi573]

Very nice!!! I solved it but unfortunately I missed the second solution!!thank you!!! Great ❤❤❤❤❤❤❤

[@SyberMath]

Thank you! 💕

[@think_logically_]

I did the same, but using negative powers is unnecessary. Once you've got (4x)^x = 1/(4^(1/16)), rewrite it as (4x)^x = (1/4)^(1/16), and raise to power 4, getting (4x)^(4x) = (1/4)^(1/4) , so 4x=1/4 and x=1/16.

[@SyberMath]

Pretty good

[@yoav613]

Nice! If you take x=2^-k you get 2^(k-3)=k-2 and k=3,4 works.

[@SyberMath]

Good!

[@JamesDavy2009]

You just defined a theorem of tetration: a^^2 = b^^2, a = b. Not sure if it holds for higher level power towers.

[@wikopl]

What is the value of blue curve when x=0? Is it defined?

[@SyberMath]

No

[@ozymandias8523]

Damn thanks for sharing

[@SyberMath]

Np

[@MartinPerez-oz1nk]

THANKS PROFESOR!!!!!, VERY INTERESTING !!!!

[@SyberMath]

Glad you liked it!

[@scottleung9587]

Thought it was case closed when I found x=1/16 - didn't think about making further manipulations after that.

[@SyberMath]

🤫😉

[@defeat7774]

You are amazing wow,👍🏻👍🏻👍🏻

[@SyberMath]

Thank you so much 😀

[@2012tulio]

Why did I get only 1/8 ? (1/x)^x = 4^x +1/16 Ln both sides Xln 1/x = (x+1/16) ln4 Xln x^-1 = xln4 + 1/16 ln4 Xln x = -xln4 - 1/16 ln4 Xlnx +x ln4 = -1/16 ln4 X(lnx+ ln4) = -1/16 ln4 Multiply by 4 both sides 4x ln 4x = -1/4 ln 4 Let 4x = y Y lny = -1/4 ln 4 W(e^lny.lny) = W(-1/4 ln 4) Lny = W (-1/4 ln 4) Y= e^W(-1/4 ln 4) Y= ½ 4x = ½ X = 1/8

[@fabriziosantin6063]

Cool!

[@-basicmaths862]

Substitute x=4^k & solving this u get k=-2.

[@SyberMath]

Nice

[@RAG981]

nice

[@novayarussia]

Don’t just make up equations for the pleasure of solving them… what if you accidentally find something intractable and fascinating?!?

[@omarqasim7516]

Great

[@mathswan1607]

x=1/16

[@havrulenkoevgeniy158]

Good)

[@Caseofgames]

I don't know what it is, but I just don't like the term 'homemade' in this context.

[@SyberMath]

Can you elaborate?

[@Caseofgames]

@@SyberMath It just feels off to me.

[@omsingh7683]

@@SyberMath he is jealous of your knowledge just ignore him sir you are genius

[@yakupbuyankara5903]

1/8.

[@Zitr_z]

Me who get e^(-1/8)

[@vladimirkaplun5774]

Your shoes a^1/a=b^1/b are wore out already. How will you find the second solution if x^1/x=18^1/18 for example?

[@SyberMath]

Eeezee! 😄😜 postimg.cc/SJbJ7Mz3

[@robertveith6383]

OP -- Each of your fractional exponents must be inside grouping symbols when written in this style because of the Order of Operations.

[@vladimirkaplun5774]

@@SyberMath 🤣

[@vladimirkaplun5774]

@@robertveith6383 Did you want to show off? Syber clearly understands what I meant.

[@rakenzarnsworld2]

x = 1/16