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This is an interesting question from Reddit AskMath. Solve for positive real values of x such that:
xx8 = 8
As usual, watch the video for a solution.
Or keep reading.
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“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.
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Answer To Equation x to power x to power 8 equals 8
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
We can graph both functions to get an idea. The exponential equation will grow at a ridiculous (exponential!) rate, whereas y = 8 is a flat time.
The graphs cross one time and then the exponential equation will continue increasing, so there is exactly one solution. So how can we find it?
Here’s the trick! Think about the equation:
x8 = 8
There is an exponent of 8. But we also know 8 is equal to the entire left-hand side, which is x8. So what if we substituted x8 for the exponent of 8? We get exactly the equation that we want to solve:
x(x8) = 8
xx8 = 8
The solution of the original equation will be a solution to the equation we want to solve. But the original equation is easy to solve in positive real numbers. We take both sides to the power 1/8 and we get:
x = 81/8 ≈ 1.297
References
Reddit AskMath
https://www.reddit.com/r/askmath/comments/17s5s97/my_friend_posted_this_and_i_really_wanna_know_how/
Excel goal seek idea u/StL_TrueBlue91
https://www.reddit.com/r/askmath/comments/17s5s97/comment/k8p7eeq/
Power tower image
https://en.wikipedia.org/wiki/File:Infinite_power_tower.svg
RicHard-59, CC BY-SA 3.0
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