The real answer to this 'Harvard' entrance exam

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The equation 3^x = x^9 is quite challenging, and some videos illustrate a trick to find one solution. But how can you find all real valued solutions? 4:05 has a typo: the value 1.30 should read 1.13. Special thanks this month to: Lee Redden, Kyle. Thanks to all supporters on Patreon and YouTube! http://www.patreon.com/mindyourdecisions https://www.youtube.com/@MindYourDecisions/join How can we solve 3^x = x^9 for real values of x? On the one hand 3 to the power of x is an exponential equation, but the other side x to the power of 9 is a polynomial equation. To find all solutions, we need to utilize a function that is not taught in schools. 0:00 problem 1:25 partial answer 4:25 surprise 6:33 lambert W 8:25 branches 12:27 practical 13:18 all solutions MrMath https://www.youtube.com/watch?v=7FdYu2ioIGY Map of mathematics https://www.youtube.com/watch?v=2A51f4yRd4w Math StackExchange https://math.stackexchange.com/questions/4867165/an-analytic-solution-to-solve-x9-3x https://math.stackexchange.com/questions/3463603/solve-for-x-in-ax-xb?noredirect=1&lq=1 original desmos (not my creation) https://www.desmos.com/calculator/wqwoud7uvk adapted desmos https://www.desmos.com/calculator/xajrvmzsaz Members get early access to videos YouTube members https://www.youtube.com/channel/UCHnj59g7jezwTy5GeL8EA_g/join Patreon members https://www.patreon.com/mindyourdecisions Support the channel with a one-time donation https://www.paypal.com/donate/?hosted_button_id=REZRY3QRQPYQA Subscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1 Send me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas! If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay. Book ratings are from January 2025. My Books (worldwide links) https://mindyourdecisions.com/blog/my-books/#worldwide My Books (US links) Mind Your Decisions: Five Book Compilation https://amzn.to/2pbJ4wR A collection of 5 books: "The Joy of Game Theory" rated 4.2/5 stars on 564 reviews http://amzn.to/1uQvA20 "The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" rated 4.2/5 stars on 55 reviews http://amzn.to/1o3FaAg "40 Paradoxes in Logic, Probability, and Game Theory" rated 4.2/5 stars on 81 reviews http://amzn.to/1LOCI4U "The Best Mental Math Tricks" rated 4.3/5 stars on 148 reviews http://amzn.to/18maAdo "Multiply Numbers By Drawing Lines" rated 4.5/5 stars on 57 reviews http://amzn.to/XRm7M4 Mind Your Puzzles: Collection Of Volumes 1 To 3 https://amzn.to/2mMdrJr A collection of 3 books: "Math Puzzles Volume 1" rated 4.4/5 stars on 138 reviews http://amzn.to/1GhUUSH "Math Puzzles Volume 2" rated 4.2/5 stars on 45 reviews http://amzn.to/1NKbyCs "Math Puzzles Volume 3" rated 4.3/5 stars on 38 reviews http://amzn.to/1NKbGlp 2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter. My Blog https://mindyourdecisions.com/blog/ Twitter https://twitter.com/preshtalwalkar Instagram https://www.instagram.com/preshtalwalkar/ Merch https://teespring.com/stores/mind-your-decisions Patreon https://www.patreon.com/mindyourdecisions Press https://mindyourdecisions.com/blog/press