If x^y=8 and x is bigger than 1, then what's the minimum of x+y=?

If x^y=8 and x is bigger than 1, find the minimum of x+y. At first, you might notice 2^3=8 and 2+3=5 and that will be the answer, however, what if x is not a whole number? This seemingly easy calculus 1 optimization problem requires the Lambert W function to find its exact value. This question is similar to my previous video https://youtu.be/zdAJXil-NvA?si=vJ95sudIGXpua6D0 More about the Lambert W function: https://youtu.be/Qb7JITsbyKs?si=-sBLaf12qel9llEZ ---------------------------------------- Big thanks to my Patrons for the full-marathon support! Ben D, Grant S, Mark M, Phillippe S. Michael Z, Jan P. Devun C. Stefan C. Ethan BW Didion S. NN Minkyu Y Brandon F Levon M shortsleeve Jack P Gabriel G Yeeted C David H Mateo F Emma M Eden E Minicat Jose R Nicholas Fey Tyler B Lucas W Ahmet Ö An R Camila L Luupo Farrah Jordan T Batman 1127 Mathguru Дмитрий П 💪 Support this channel and get my math notes by becoming a patron: https://www.patreon.com/blackpenredpen 🛍 Shop my math t-shirt & hoodies: https://amzn.to/3qBeuw6 ---------------------------------------- #blackpenredpen #math #calculus #apcalculus