A Nice Olympiads Exponential Mathematics | How to Solve for k?

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[@mathswan1607]

kln 2=32lnk lnk/k=ln2/32=8ln2/256=ln256/256 Hence k=256

[@alphalunamare]

I find this an unsatisfactory dive into the unknown. I like 1:55 but my instinct would be to take logs base 2 and then have: K = 32*L2(K) Clearly K is a multiple of 32 and inspection quickly leads to the answer. No need for all that garbage Lambert functional complication in other posts.

[@MathSync]

Nice Explanation❤

[@anthonyvalenti9093]

Actually 3 solutions 256 1.02239 -.979017

[@aarinbishes3002]

Already seen this format in many other videos..Ans 256

[@aminimam5118]

nice, but if you use the Lambert W function, you will have a second solution 1.022393

[@erwinkurniadi1850]

Is there a short way to convert 2^1/32 to 256^1/256?

[@choseongho32]

You had to restrict k to be a positive integer.

[@prollysine]

by W Lambert , k*ln2=32*lnk , ln2/32=k^(-1)*lnk , ln2/32=lnk*e^(ln(-k)) , ln2/32=lnk*e^(-lnk) ,*(-1) . W(-ln2/32)=W(- lnk*e^(-lnk)) , -lnk=W(ln2/32) , lnk=-W(-ln2/32) , W(-ln2/32)=~ -0.022145899 , lnk=~ 0.022145899 , k=e^0.022145899 , k=~ 1.0223929396969 , test , 2^1.0223929396969=~ 2.03129 , 1.0223929396969^32=~ 2.03129 , same , OK ,