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

  • Context: MHB 
  • Thread starter Thread starter DrLiangMath
  • Start date Start date
  • Tags Tags
    Exponential
Click For Summary
SUMMARY

The equation \(4^x + 3^x = 12\) does not have an exact solution, as confirmed by multiple participants in the forum discussion. Attempts to solve it using various methods, including numerical approximations and comparisons to similar equations, yielded only approximate results. Tools like Wolfram Alpha also provide no exact solutions, reinforcing the consensus that the equation remains unsolvable in closed form. The discussion highlights the challenges of finding exact solutions for exponential equations of this nature.

PREREQUISITES
  • Understanding of exponential equations
  • Familiarity with numerical approximation methods
  • Basic knowledge of mathematical software tools like Wolfram Alpha
  • Experience with algebraic manipulation and equation solving techniques
NEXT STEPS
  • Research methods for solving exponential equations
  • Explore numerical approximation techniques in mathematics
  • Learn about the limitations of Computer Algebra Systems (CAS) in solving complex equations
  • Investigate similar equations, such as \(4^x + 6^x = 9^x\), for comparative analysis
USEFUL FOR

Mathematicians, students studying algebra, and anyone interested in the complexities of solving exponential equations will benefit from this discussion.

DrLiangMath
Messages
21
Reaction score
0
Solve $4^x+3^x=12$ without using CAS?
 
Mathematics news on Phys.org
graph.JPG
 
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.
 
  • Like
Likes   Reactions: SammyS

Similar threads

MHB Solve Equation: $x^4+2x^3-x^2-6x-3=0$
  • · Replies 7 ·
Replies
7
Views
2K
MHB -s6.6 solve exponential eq 3^x-14\cdot 3^{-x}=5
  • · Replies 1 ·
Replies
1
Views
1K
MHB What is the Solution for $5^x+2=12$?
  • · Replies 4 ·
Replies
4
Views
1K
MHB Solve $x^4+(4-x)^4=32$ Equation
  • · Replies 2 ·
Replies
2
Views
1K
MHB Calculating Integral with Simpson's Rule for Error < $0.5\cdot 10^{-3}$
  • · Replies 6 ·
Replies
6
Views
2K
MHB No Real Zeroes of $x^6-x^5+x^4-x^3+x^2-x+\dfrac{3}{4}$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Solving a Polynomial x^6 – 7x^3 + 12 by Factoring.
  • · Replies 2 ·
Replies
2
Views
2K
MHB 6.2.8 {{k,4},{3,k}}=k what is k
  • · Replies 3 ·
Replies
3
Views
1K
MHB Solve System of 4 Equalities: x+y+z+w=22, xyzw=648
  • · Replies 6 ·
Replies
6
Views
2K
MHB Linear combination of sine and cosine function
  • · Replies 3 ·
Replies
3
Views
2K