Solve For { X $}$ In The Equation: ${ \ln 6 X^2 - 24 X = 0 }$
Introduction to the Problem
Understanding the Equation The given equation is a logarithmic equation involving a natural logarithm. The equation is ${ \ln 6 x^2 - 24 x = 0 }$. To solve for , we need to isolate the variable and find its value.
Step 1: Isolate the Logarithmic Term
The first step is to isolate the logarithmic term on one side of the equation. We can do this by adding to both sides of the equation:
{ \ln 6 x^2 - 24 x + 24 x = 24 x \}$
${ \ln 6 x^2 = 24 x }$
To eliminate the logarithm, we can exponentiate both sides of the equation using the base of the natural logarithm, which is . This will give us: { e^{\ln 6 x^2} = e^{24 x} \}$
${ 6 x^2 = e^{24 x} }$
## Step 3: Simplify the Equation
Since $e^{\ln x} = x$, we can simplify the equation to:
${ 6 x^2 = e^{24 x} \}$
${ 6 x^2 = (e^2)^{12 x} \}$
${ 6 x^2 = (e^2)^{12 x} \}$
## Step 4: Take the Square Root of Both Sides
To get rid of the square root, we can take the square root of both sides of the equation:
${ \sqrt{6 x^2} = \sqrt{(e^2)^{12 x}} \}$
${ \sqrt{6} x = (e^2)^{6 x} \}$
## Step 5: Simplify the Equation
Since $\sqrt{6} = \sqrt{2} \cdot \sqrt{3}$, we can simplify the equation to:
${ \sqrt{2} \cdot \sqrt{3} x = (e^2)^{6 x} \}$
${ \sqrt{2} x \sqrt{3} = (e^2)^{6 x} \}$
## Step 6: Divide Both Sides by $\sqrt{3}$
To isolate $x$, we can divide both sides of the equation by $\sqrt{3}$:
${ \frac{\sqrt{2} x \sqrt{3}}{\sqrt{3}} = \frac{(e^2)^{6 x}}{\sqrt{3}} \}$
${ \sqrt{2} x = \frac{(e^2)^{6 x}}{\sqrt{3}} \}$
## Step 7: Divide Both Sides by $\sqrt{2}$
To finally isolate $x$, we can divide both sides of the equation by $\sqrt{2}$:
${ \frac{\sqrt{2} x}{\sqrt{2}} = \frac{\frac{(e^2)^{6 x}}{\sqrt{3}}}{\sqrt{2}} \}$
${ x = \frac{(e^2)^{6 x}}{\sqrt{2} \cdot \sqrt{3}} \}$
${ x = \frac{(e^2)^{6 x}}{\sqrt{6}} \}$
## Conclusion
The final solution to the equation ${ \ln 6 x^2 - 24 x = 0 }$ is $x = \frac{(e^2)^{6 x}}{\sqrt{6}}$. This is a complex solution that involves the use of logarithmic and exponential functions.
## Final Answer
The final answer is: $\boxed{\frac{(e^2)^{6 x}}{\sqrt{6}}}{{content}}amp;lt;br/>
# Q&A: Solving the Equation ${ \ln 6 x^2 - 24 x = 0 }$
## Frequently Asked Questions
### Q: What is the main concept behind solving the equation ${ \ln 6 x^2 - 24 x = 0 }$?
A: The main concept behind solving this equation is to isolate the variable $x$ and find its value. This involves using logarithmic and exponential functions to simplify the equation.
### Q: How do I start solving the equation ${ \ln 6 x^2 - 24 x = 0 }$?
A: To start solving the equation, you need to isolate the logarithmic term on one side of the equation. This can be done by adding $24x$ to both sides of the equation.
### Q: What is the next step after isolating the logarithmic term?
A: After isolating the logarithmic term, you need to exponentiate both sides of the equation using the base of the natural logarithm, which is $e$. This will eliminate the logarithm and simplify the equation.
### Q: How do I simplify the equation after exponentiating both sides?
A: After exponentiating both sides, you can simplify the equation by using the property of logarithms that states $e^{\ln x} = x$. This will give you a simplified equation that is easier to work with.
### Q: What is the final step in solving the equation ${ \ln 6 x^2 - 24 x = 0 }$?
A: The final step in solving the equation is to isolate $x$ by dividing both sides of the equation by $\sqrt{6}$. This will give you the final solution to the equation.
### Q: What is the final solution to the equation ${ \ln 6 x^2 - 24 x = 0 }$?
A: The final solution to the equation is $x = \frac{(e^2)^{6 x}}{\sqrt{6}}$. This is a complex solution that involves the use of logarithmic and exponential functions.
### Q: Why is the solution to the equation ${ \ln 6 x^2 - 24 x = 0 }$ complex?
A: The solution to the equation is complex because it involves the use of logarithmic and exponential functions. These functions can be difficult to work with, especially when they are combined in complex ways.
### Q: What are some common mistakes to avoid when solving the equation ${ \ln 6 x^2 - 24 x = 0 }$?
A: Some common mistakes to avoid when solving this equation include:
* Not isolating the logarithmic term on one side of the equation
* Not exponentiating both sides of the equation
* Not simplifying the equation after exponentiating both sides
* Not isolating $x$ by dividing both sides of the equation by $\sqrt{6}$
### Q: How can I practice solving equations like ${ \ln 6 x^2 - 24 x = 0 }$?
A: You can practice solving equations like this by working through example problems and exercises. You can also try solving different types of equations, such as linear and quadratic equations, to build your skills and confidence.
## Additional Resources
* For more information on solving logarithmic and exponential equations, see the following resources:
+ Khan Academy: Logarithmic and Exponential Equations
+ Mathway: Logarithmic and Exponential Equations
+ Wolfram Alpha: Logarithmic and Exponential Equations
## Conclusion
Solving the equation ${ \ln 6 x^2 - 24 x = 0 }$ requires a combination of logarithmic and exponential functions. By following the steps outlined in this article, you can isolate the variable $x$ and find its value. Remember to avoid common mistakes and practice solving different types of equations to build your skills and confidence.</span></p>
Step 2: Exponentiate Both Sides