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 should read . Special thanks this month to: Lee Redden, Kyle. Thanks to all supporters on Patreon and YouTube! @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 Map of mathematics Math StackExchange original desmos (not my creation) adapted desmos Members get early access to videos YouTube members Patreon members Support the channel with a one-time donation Subscribe: 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) #worldwide My Books (US links) Mind Your Decisions: Five Book Compilation A collection of 5 books: "The Joy of Game Theory" rated 4.2/5 stars on 564 reviews "The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" rated 4.2/5 stars on 55 reviews "40 Paradoxes in Logic, Probability, and Game Theory" rated 4.2/5 stars on 81 reviews "The Best Mental Math Tricks" rated 4.3/5 stars on 148 reviews "Multiply Numbers By Drawing Lines" rated 4.5/5 stars on 57 reviews Mind Your Puzzles: Collection Of Volumes 1 To 3 A collection of 3 books: "Math Puzzles Volume 1" rated 4.4/5 stars on 138 reviews "Math Puzzles Volume 2" rated 4.2/5 stars on 45 reviews "Math Puzzles Volume 3" rated 4.3/5 stars on 38 reviews 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 Twitter Instagram Merch Patreon Press