MHB Is There an Exact Solution to the Equation \(4^x + 3^x = 12\)?

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The equation \(4^x + 3^x = 12\) does not have an exact solution, as attempts to solve it using various methods yield only approximate results. Participants in the discussion express skepticism about claims of an exact solution, with one noting a similar experience where they received only numerical approximations from computational tools like Wolfram Alpha. The conversation reflects frustration over the lack of a clear method to derive an exact solution, with references to other equations that can be solved more easily. Overall, the consensus is that the equation remains unsolvable in exact terms. The discussion highlights the challenges of finding precise solutions for exponential equations.
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Solve $4^x+3^x=12$ without using CAS?
 
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maxkor said:
View attachment 11883
Nice solution! But if you could find an exact solution it would be much better.
 
MathTutoringByDrLiang said:
Nice solution! But if you could find an exact solution it would be much better.
Do you have an answer?
 
maxkor said:
Do you have an answer?

I don't have such solution. A person put some comment on my YouTube video and ask me to solve this equation. I tried using a similar method as you, and got the same approximate solution. Then he said he could get an exact solution but never showed me the solution. I was just wondering if he actually put a joke on me.
 
MathTutoringByDrLiang said:
I don't have such solution. A person put some comment on my YouTube video and ask me to solve this equation. I tried using a similar method as you, and got the same approximate solution. Then he said he could get an exact solution but never showed me the solution. I was just wondering if he actually put a joke on me.
I can solve something like $4^x + 6^x = 9^x$ only because I can put it into a homogeneous format. I can't think of any way to do this with $3^x + 4^x = 12$. Even if there were a typo I don't think we can do $3^x + 4^x = 12^x$.

-Dan
 
If you plug it into wolfram alpha, I don't think you can solve this with a CAS either, it's just numeric approximations for you.
 
DrLiangMath said:
I don't have such solution. A person put some comment on my YouTube video and ask me to solve this equation. I tried using a similar method as you, and got the same approximate solution. Then he said he could get an exact solution but never showed me the solution. I was just wondering if he actually put a joke on me.

Reminds me of...
Fermat said:
I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.
 
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