Solving 3^x=12x-9 Algebraically: Step-by-Step Guide

  • Thread starter physicsdreams
  • Start date
In summary, the equation 3^x=12x-9 has two solutions, x=1 and x=3. The first solution can be found by graphing the two functions and finding their points of intersection, while the second solution involves using the Lambert W-function, which is typically learned in postcalculus courses. It is also possible to solve this algebraically by manipulating the equation into the form 3^{x-2} + 1 = \frac{4x}{3}, but this method may not always yield the correct solution.
  • #1
physicsdreams
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0

Homework Statement



3^x=12x-9

Homework Equations





The Attempt at a Solution



I really have no clue how to solve this one algebraically.
I graphed the two functions on a calculator and found the points of intersection
the answers are 3 and 1

Can someone show me how to solve this problem algebraically, step by step?

I believe you use logs?

thanks
 
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  • #2


physicsdreams said:
Can someone show me how to solve this problem algebraically, step by step?
No, we're not allowed to do that. That's against forum rules.

I'm not sure it can be solved algebraically. The solution provided by Wolfram used the Lambert W-function, but I have to confess that I don't remember what that is.
 
  • #3


[tex]3^x=3(4x-3)[/tex]
When does the left side equal the right side?
 
  • #4


eumyang said:
No, we're not allowed to do that. That's against forum rules.

I'm not sure it can be solved algebraically. The solution provided by Wolfram used the Lambert W-function, but I have to confess that I don't remember what that is.

The function f(x)=xe^x is a one-to-one function if you restrict the domain to x in [-1,infinity). So it has an inverse. The inverse is the Lambert W-function. It's just a name for a function you can't express algebraically.
 
  • #5


thanks guys for the help,
sorry about asking for the step by step, I'm new here.

as for the Lambert w-function, is it something one learns in Calculus or something?
I'm only in precalculus.
 
  • #6


physicsdreams said:
thanks guys for the help,
sorry about asking for the step by step, I'm new here.

as for the Lambert w-function, is it something one learns in Calculus or something?
I'm only in precalculus.

I think I learned about it in a course called Mathematical Physics which is postcalculus, where you study all sorts of special functions associated with differential equations. I don't think are supposed to solve the equation like that. Do just what you did. Sketch the graph and guess the roots.
 
  • #7


You can solve this algebraically.
 
  • #8


[tex]
3^x = 3(4x-3)
[/tex]

[tex]3^{x-1} = 4x-3[/tex]
[tex]3^{x-1} + 3 = 4x[/tex]
[tex]3^{x-2} + 1 = \frac{4x}{3}[/tex]
[tex]\frac{3^{x-2}}{x} + \frac{1}{x} = \frac{4}{3}[/tex]

See if you can go from there.
 
  • #9


mjordan2nd said:
[tex]
3^x = 3(4x-3)
[/tex]

[tex]3^{x-1} = 4x-3[/tex]
[tex]3^{x-1} + 3 = 4x[/tex]
[tex]3^{x-2} + 1 = \frac{4x}{3}[/tex]
[tex]\frac{3^{x-2}}{x} + \frac{1}{x} = \frac{4}{3}[/tex]

See if you can go from there.
And this is helpful...how?
 
  • #10


Mark44 said:
And this is helpful...how?

I am assuming that it is, in fact, impossible to solve algebraically?
I really don't see how moving everything around gets me closer to an answer.

Thanks for trying though, mjordan2nd, unless there really is a way to solve it by taking your route. Please feel free to give me a few more hints as to how this helps.

(I'm still learning here!)

Thanks
 
  • #11


Mark44 said:
And this is helpful...how?

Ahh, I suppose you're right. My thinking was you could simply equate the top with the top and the bottom with the bottom. In this case equating the bottom with the bottom gives x = 3, which happens to work in this case, but doesn't have to. My fault.
 

1. How do I solve 3^x=12x-9 algebraically?

To solve this equation algebraically, you can use the logarithm property of equality, which states that if logb(x) = logb(y), then x = y. In this case, we can take the logarithm of both sides of the equation to get log3(3^x) = log3(12x-9). This simplifies to x = log3(12x-9). From here, you can use trial and error or a calculator to find the approximate value of x.

2. Can I use any base for the logarithm in this equation?

Yes, you can use any base for the logarithm in this equation. However, it is recommended to use the same base on both sides of the equation to simplify the calculation process.

3. What are the possible solutions for this equation?

This equation has an infinite number of solutions, as there are an infinite number of values for x that can satisfy the equation. However, if you are looking for a specific solution, you can use a calculator or numerical methods to find an approximate value.

4. Is there a specific method to solve this equation?

There are several methods you can use to solve this equation, such as using logarithms, substitution, or graphing. The method you choose may depend on your personal preference or the complexity of the equation.

5. Can I solve this equation without using logarithms?

Yes, you can solve this equation without using logarithms. As mentioned earlier, there are other methods you can use, such as substitution or graphing, to find a solution. However, using logarithms is usually the most efficient method for solving exponential equations like this one.

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