Algebraic Solution for x + 3^x = 4

  • Thread starter zoxee
  • Start date
In summary, The conversation discusses how to solve the equation x + 3^x = 4 using algebraic methods. The solution of x = 1 can be found by visually inspecting the equation, but the conversation also suggests using the Lambert W function. However, it is mentioned that it is not a purely algebraic method and instead, setting f(x) = x+3^x-4 and testing for monotonicity can prove that there are no other solutions. Ultimately, it is concluded that solving the equation algebraically is not possible.
  • #1
zoxee
37
0
how can I solve this using algebraic methods? I know the solution is x = 1 from just looking at it, but not sure how to do it algebraically:

## x + 3^x = 4 ##
 
Physics news on Phys.org
  • #3
You can set f(x) = x+3^x-4, take x=1 as a solution to f(x) = 0 and look if it is monotonic to show that it has no other solutions. Algebraically is not possible.
 
  • #4
zoxee said:
how can I solve this using algebraic methods? I know the solution is x = 1 from just looking at it, but not sure how to do it algebraically:
## x + 3^x = 4 ##
a very strange way of writing one:

[tex]x=\frac{-W(3^4\ln(3))}{\ln(3)}+4=1[/tex]
 

1. What is an algebraic solution?

An algebraic solution is a method of solving an equation or problem using algebraic techniques, such as manipulating symbols and equations to find a specific value or set of values.

2. How do you solve x + 3^x = 4?

To solve this equation, you can use the algebraic technique of isolating the variable on one side of the equation. First, subtract 4 from both sides to get x + 3^x - 4 = 0. Then, you can use logarithms to rewrite the equation as x + log3(3^x) - log3(4) = 0. Simplifying further, you get x + xlog3(3) - log3(4) = 0. Since log3(3) = 1, this simplifies to 2x - log3(4) = 0. Finally, solve for x by dividing both sides by 2 and using the change of base formula to rewrite log3(4) as log(4)/log(3). The final solution is x = log(4)/2log(3).

3. Can this equation be solved without using logarithms?

Yes, there is an alternate method for solving this equation without using logarithms. You can use the substitution method by letting y = 3^x. This will change the equation to y + x = 4. Then, you can use the quadratic formula to solve for y, which will give you two solutions. Finally, you can plug these solutions back into the original equation to solve for x.

4. Are there any restrictions on the values of x for this equation?

Yes, since logarithms are only defined for positive numbers, the solution x = log(4)/2log(3) is only valid for positive values of 3^x. This means that x must be greater than 0 for this solution to be valid.

5. Can this equation be solved graphically?

Yes, you can use a calculator or graphing software to plot the two sides of the equation separately and find the point(s) of intersection. However, this method may not give an exact solution and may be less accurate than using algebraic techniques.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
8
Views
134
  • Precalculus Mathematics Homework Help
Replies
2
Views
709
  • Precalculus Mathematics Homework Help
Replies
7
Views
1K
  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
11
Views
1K
  • Precalculus Mathematics Homework Help
Replies
8
Views
292
  • Precalculus Mathematics Homework Help
Replies
23
Views
433
  • Precalculus Mathematics Homework Help
Replies
19
Views
1K
  • Precalculus Mathematics Homework Help
Replies
21
Views
432
  • Precalculus Mathematics Homework Help
Replies
5
Views
705
Back
Top