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

[DrLiangMath]

Solve $4^x+3^x=12$ without using CAS?

 

[maxkor]

 

[DrLiangMath]

View attachment 11883

Nice solution! But if you could find an exact solution it would be much better.

 

[maxkor]

Nice solution! But if you could find an exact solution it would be much better.

Do you have an answer?

 

[DrLiangMath]

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.

 

[topsquark]

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

 

[Office_Shredder]

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.

 

[robphy]

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...

I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.