An Interesting Exponential Equation

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[dorkmania]

The last approach for W((-2/9)•ln(3)) is the same as W((-1/9)•ln(9)) which gives the same solution as before, not 1.18815... necessarily.

[fahrenheit]

Ooh, looks nice. As usual, here's my guess prior to watching.
Given the powers of 3 present, I figured it would be a good idea to write x = 3^k, noting that x must be positive (if real) to ensure the RHS is positive.
From this, we see that 3^27k = (3^3)^(3^2k) = (3^3)^(9k) so 3^2k = 9k, so 9^k = 9k, which has 2 solutions, k=1, and some smaller one.
I think perhaps k = -W(-ln(9)/9)/ln(9) with W being the Lambert W function, but I might have messed that up, and that possibly provides the second solution, using another branch?. The actual values of x are therefore 3, and some smaller root, still greater than 1.

[mcwulf]

I knew there would be a W. Left it to the end!

[michaelwilson]

👎 Lambert's bloody W function! Hate