Logarithmic Equation with x on both sides

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Homework Help Overview

The problem involves an exponential equation with the variable x present on both sides, specifically the equation 18x^2=6e^(2x). Participants are exploring methods to manipulate and solve this equation, particularly through logarithmic transformations.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss taking logarithms of both sides and question the correctness of the manipulations. There is a focus on whether the equation can be solved using elementary functions or if numerical methods are required.

Discussion Status

Some participants have provided feedback on the original poster's notation and suggested that the equation cannot be solved analytically. There is an ongoing exploration of numerical methods, including the use of the intermediate value theorem to demonstrate the existence of solutions.

Contextual Notes

There is a noted concern regarding the use of parentheses in the original equation, which may affect the interpretation of the mathematical expressions. The discussion also reflects uncertainty about the methods available for solving the equation.

Negima
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Exponential Equation with x on both sides

Homework Statement



18x^2=6e^(2x)

Homework Equations


I don't know any for the step I got stuck on.


The Attempt at a Solution


3x^2=e^2x
ln3x^2=lne^2x
2ln3x=2x
ln3x=x
 
Last edited:
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Please start using more parentheses, ok? If you mean 3*x^2=e^(2x), then taking logs gives you log(3)+2*log(x)=2*x. The left side is not 2ln3x, whatever that means. You can't solve that using elementary functions. You can prove a solution exists, and you can solve it numerically, but that's about it.
 
Thank you for your reply! And sorry about the lack of parenthesis.

By solving it numerically, do you mean guess and checking values for x? Or actually solving for x?
 
Negima said:
Thank you for your reply! And sorry about the lack of parenthesis.

By solving it numerically, do you mean guess and checking values for x? Or actually solving for x?

Well yes you can guess values using the intermediate value theorem and show that a solution exists.
 

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