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So basically I’m trying find the inverse to this formula and I’m having trouble getting it could anyone help me out I would appreciate it thank you

$$f_{3t}(x) = \frac{\log(25-(-1.8(x)))}{\log(x)}$$

edit: to clarify I am under the assumption that the inverse currently does not exist as I was trying to make a formula to solve this polynomial (x^n-1.8x=25) once I had that solved I could change up the numbers to eventually find a formula for (x^n+ax) I had worked on this for about 3 weeks and while I did manage to obtain the inverse to n I need the inverse of that equation to solve for it hence the original question I have only lightly skimmed through your answers but I will go back and check some of the links you provided me hopefully together we can find the inverse formula for this equation I also didn’t mention that it was currently unsolved as 1 I didn’t know if that’s true and 2 I didn’t want to discourage anyone from trying hopefully that answers the reason behind the question if you have anything else you want me to explain like the work I did to get that formula or other feel free to let me know! P.S sorry for the lack of punctuation I’m just a bit excited I can finally have others to have a look at it.

edit#2: here is what the inverse should look like image of the inverse and here was the closest I got to finding it my attempts to approximate the inverse hope all of this helps!

edit#3: here is all the work I’ve done to get the formula sorry in advance for all the disorganization https://www.desmos.com/calculator/slkoghiext https://www.desmos.com/calculator/t4qt5rvsp5 https://www.desmos.com/calculator/tvxtd6fvzu

https://www.desmos.com/calculator/gnrfbz0awz

https://www.desmos.com/calculator/9lpaor8brm there’s more but those are either not important or completely failed as stated before hope this helps!

SelfProclaimedDev
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    I'm quite sure that function does not have a closed form algebraic solution. I think you will need numerical methods. If you [edit] the question to tell us where the function comes from and why you are inverting it we might be able to help. – Ethan Bolker Jun 09 '22 at 13:15
  • This might be useful. – Number Basher Jun 09 '22 at 13:23
  • Ah this is referring to the lambert W thingy I’ve heard about it and tried to use it but I ultimately confused myself when trying to implement it to the formula which was my final attempt at doing it mostly on my own so I decided to come here and share the information to see if anyone else could figure it out. – SelfProclaimedDev Jun 09 '22 at 16:03
  • There is no formula for the solution to the equation $x^n-1.8x=25$ for general $n$. I don't think many people here will look at all your desmos attempts. – Ethan Bolker Jun 09 '22 at 16:06
  • @EthanBolker I understand that but seeing how I was able to get the inverse of the formula for the solution I am now confident that it does exist maybe in the form of a summation or exponential point being I don’t wanna give up on it just yet but thanks for confirming that there is no formula to solve this as of yet! – SelfProclaimedDev Jun 09 '22 at 16:09
  • You said "inverse currently does not exist", but it does exist, it's just, say, ugly. – Number Basher Jun 10 '22 at 00:13
  • @NobodyNeedsNames I don’t fear that do tell me how to find it. – SelfProclaimedDev Jun 10 '22 at 00:19
  • Maybe I should make it clearer. There isn't a way to express the inverse (except writing $ f^{-1}(x) $ probably), but you can get an approximation, which you have done. – Number Basher Jun 10 '22 at 00:24
  • @NobodyNeedsNames my goal is to narrow down the approximate to be as accurate as possible then simplify it does make the formula bigger but it works pretty well – SelfProclaimedDev Jun 10 '22 at 00:28
  • OK IHATE!%CHAR!!! – Number Basher Jun 10 '22 at 00:30
  • @NobodyNeedsNames on a plus note I did find the exact formula for when y = 0 – SelfProclaimedDev Jun 10 '22 at 00:36
  • Let me think about this problem different if we have the formula and know what the inverse looks like we can just plot all of those digets on a table then have a script that just swaps the numbers reorders them from least to greatest and finally plot out the data table do this enough times it should give a pretty effective inverse – SelfProclaimedDev Jun 10 '22 at 01:39
  • Let us continue this discussion in chat. – Number Basher Jun 10 '22 at 01:54

1 Answers1

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According to this answer, there is no definite inverse. Are you asking for an approximation? What is the specific question of the, say, homework, or did you think of this question up? You might helpful sources under if you search log. Usually writing "basically" as the reason of why you want to solve a problem isn't enough or even useful. (Also note that I am not smart enough to come up with an approximation, as I am only a seventh-grader).

Your function $f$ is an open bijection, $f^{-1}$ exists, but does not have an easy analytical expression.

Number Basher
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    As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Jun 09 '22 at 14:35
  • Btw I’m very new to posting questions so please excuse me for any bad formatting or mistakes I’ll try to fix them as I go so feel free to point out anything I might of missed thanks! – SelfProclaimedDev Jun 09 '22 at 15:44