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What would the method be to multiply logarithms? Is that possible?

Background: My professor assigned us a few problems to be worked on over half a month or so. One of these involved algebraically solving for a variable using a given equation in exact form.

I’ve changed the constants and order around a bit, so it may not be possible. Please point that out if that happens. Here’s that modified problem:

$(\log _{6}{x})(4 \log _{6}(x+2))=8$

Am I doing something wrong? I keep trying to simplify, and I end up with 8 divided by $\log _{6} (x)$ and I run into a wall there. I also tried rewriting the 8 (its equivalent in the original problem, not the number) in the log form with the same base. From there, can I just remove the log from everywhere?

Thanks so much in advance!

Edit: It appears the commenters are finding this modified problem not to work, I’m really sorry about that! Take the actual equation with a grain of salt, I’m just needing help with the method with which to work on the problem. I don’t need this equation solved. (Am I making sense?)

Edit 2: I figured out how to do it! I will edit the question with the original equation and then add an answer with what I did.

TheOneAndOnly
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    Hint: $a^{bc} = (a^b)^c$ – Dan Nov 28 '22 at 23:10
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    FWIW, Wolfram Alpha fails to find a closed-form solution, only the numerical solution $x \approx 11.65809893337614$. – Dan Nov 28 '22 at 23:16
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    I don't understand how the above hint is supposed to help with anything, since this isn't solvable. Your best bet would be W function or some generalization if you could cleverly detangle the logs, but it's not obvious to me that even that is possible. – Brevan Ellefsen Nov 28 '22 at 23:18
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    There's a chance for an infinite series representation via power series inversion, which could potentially be made numerically stable, but I'm fairly certain that's not what you're looking for given the body of the post. At an elementary level, your question is unsolvable. – Brevan Ellefsen Nov 28 '22 at 23:21
  • Either your teacher screwed up and it cannot be easily solved, or there is some trick that can be used for the specific question your teacher set. – Adam Rubinson Nov 28 '22 at 23:36
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    My guess is that the true problem had a solution you could find by inspection. this one does not. – lulu Nov 28 '22 at 23:54
  • @Dan.- There is something to clarify in this post: Wolfram gives that value because you have take the $4$ as a simple factor simplified to $2$ by division of $8$. Take instead $\log_6((x+2)^4)$ and you will get another value for $x$. Why? – Piquito Nov 29 '22 at 00:35
  • taking $6^y$ on both sides gives: $x^{4\log_6(x+2)}=6^8$. Maybe use the limit definition of the log? – Тyma Gaidash Nov 29 '22 at 13:22

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