Solve The Equation:${ 2 \ln (4x + 3) = \frac{1}{2} \ln (-4x) + 3 }$
Introduction
In this article, we will delve into solving a complex logarithmic equation. The given equation is 2 ln(4x + 3) = 1/2 ln(-4x) + 3. We will use various mathematical techniques to simplify and solve for the variable x. This equation involves logarithmic functions, which can be challenging to work with, but with the right approach, we can find the solution.
Understanding the Equation
The given equation is a logarithmic equation, which involves the natural logarithm (ln). The equation is 2 ln(4x + 3) = 1/2 ln(-4x) + 3. To solve this equation, we need to isolate the variable x. We can start by using the properties of logarithms to simplify the equation.
Using Properties of Logarithms
One of the properties of logarithms is that the logarithm of a product is equal to the sum of the logarithms. This property can be expressed as: ln(ab) = ln(a) + ln(b). We can use this property to simplify the equation.
Simplifying the Equation
Using the property of logarithms, we can rewrite the equation as:
2 ln(4x + 3) = ln((-4x)^1/2) + 3
We can simplify the right-hand side of the equation by using the property of logarithms: ln(a^b) = b ln(a). This gives us:
2 ln(4x + 3) = ln((-4x)^(1/2)) + 3
Exponentiating Both Sides
To get rid of the logarithms, we can exponentiate both sides of the equation. Since the base of the logarithm is e, we can use the property of exponents: e^ln(x) = x. This gives us:
e^(2 ln(4x + 3)) = e(ln((-4x)(1/2)) + 3)
Simplifying the Exponents
Using the property of exponents, we can simplify the left-hand side of the equation:
(e^ln(4x + 3))^2 = e(ln((-4x)(1/2)) + 3)
This gives us:
(4x + 3)^2 = e(ln((-4x)(1/2)) + 3)
Simplifying the Right-Hand Side
Using the property of exponents, we can simplify the right-hand side of the equation:
(4x + 3)^2 = (-4x)^(1/2) * e^3
Simplifying the Equation
We can simplify the equation by multiplying both sides by 2:
2(4x + 3)^2 = 2(-4x)^(1/2) * e^3
This gives us:
8x^2 + 24x + 18 = -2(4x)^(1/2) * e^3
Simplifying the Equation
We can simplify the equation by multiplying both sides by 2:
16x^2 + 48x + 36 = -4(4x)^(1/2) * e^3
This gives us:
16x^2 + 48x + 36 = -4(4x)^(1/2) * e^3
Solving for x
To solve for x, we can use the quadratic formula. The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 16, b = 48, and c = 36 - 4(4x)^(1/2) * e^3. Plugging these values into the quadratic formula, we get:
x = (-(48) ± √((48)^2 - 4(16)(36 - 4(4x)^(1/2) * e^3))) / 2(16)
Simplifying the Quadratic Formula
We can simplify the quadratic formula by plugging in the values:
x = (-48 ± √(2304 - 2304 + 256(4x)^(1/2) * e^3)) / 32
This gives us:
x = (-48 ± √(256(4x)^(1/2) * e^3)) / 32
Simplifying the Quadratic Formula
We can simplify the quadratic formula by plugging in the values:
x = (-48 ± 16(4x)^(1/2) * e^3) / 32
This gives us:
x = (-3 ± (4x)^(1/2) * e^3) / 2
Solving for x
To solve for x, we can set the two expressions equal to each other:
(-3 ± (4x)^(1/2) * e^3) / 2 = 0
This gives us:
-3 ± (4x)^(1/2) * e^3 = 0
Solving for x
We can solve for x by isolating the variable:
(4x)^(1/2) * e^3 = 3
This gives us:
(4x)^(1/2) = 3/e^3
Solving for x
We can solve for x by squaring both sides of the equation:
4x = (3/e3)2
This gives us:
4x = 9/e^6
Solving for x
We can solve for x by dividing both sides of the equation by 4:
x = 9/(4e^6)
This gives us the solution to the equation.
Conclusion
In this article, we have solved a complex logarithmic equation using various mathematical techniques. We have used the properties of logarithms to simplify the equation, exponentiated both sides to get rid of the logarithms, and used the quadratic formula to solve for the variable x. The solution to the equation is x = 9/(4e^6). This equation is a great example of how logarithmic functions can be used to model real-world problems and how mathematical techniques can be used to solve complex equations.
Final Answer
The final answer is x = 9/(4e^6).
Introduction
In our previous article, we solved the complex logarithmic equation 2 ln(4x + 3) = 1/2 ln(-4x) + 3. In this article, we will answer some of the most frequently asked questions about solving this equation.
Q: What is the main concept behind solving this equation?
A: The main concept behind solving this equation is using the properties of logarithms to simplify the equation, exponentiating both sides to get rid of the logarithms, and using the quadratic formula to solve for the variable x.
Q: What are some common mistakes to avoid when solving this equation?
A: Some common mistakes to avoid when solving this equation include:
- Not using the properties of logarithms to simplify the equation
- Not exponentiating both sides to get rid of the logarithms
- Not using the quadratic formula to solve for the variable x
- Not checking the domain of the logarithmic functions
Q: How do I check the domain of the logarithmic functions?
A: To check the domain of the logarithmic functions, you need to make sure that the argument of the logarithm is positive. In this case, the argument of the logarithm is 4x + 3 and -4x. You need to make sure that both 4x + 3 and -4x are positive.
Q: What is the solution to the equation?
A: The solution to the equation is x = 9/(4e^6).
Q: How do I verify the solution?
A: To verify the solution, you can plug the value of x back into the original equation and check if it is true.
Q: What are some real-world applications of this equation?
A: This equation has many real-world applications, including:
- Modeling population growth
- Modeling chemical reactions
- Modeling electrical circuits
Q: How do I apply this equation to real-world problems?
A: To apply this equation to real-world problems, you need to:
- Identify the variables and parameters in the problem
- Substitute the values of the variables and parameters into the equation
- Solve for the unknown variable
- Interpret the results in the context of the problem
Q: What are some common challenges when solving this equation?
A: Some common challenges when solving this equation include:
- Difficulty in simplifying the equation
- Difficulty in exponentiating both sides
- Difficulty in using the quadratic formula
- Difficulty in checking the domain of the logarithmic functions
Q: How do I overcome these challenges?
A: To overcome these challenges, you need to:
- Practice simplifying logarithmic equations
- Practice exponentiating both sides of logarithmic equations
- Practice using the quadratic formula
- Practice checking the domain of logarithmic functions
Conclusion
In this article, we have answered some of the most frequently asked questions about solving the equation 2 ln(4x + 3) = 1/2 ln(-4x) + 3. We have discussed the main concept behind solving this equation, common mistakes to avoid, how to check the domain of the logarithmic functions, the solution to the equation, how to verify the solution, real-world applications of this equation, and common challenges when solving this equation. We hope that this article has been helpful in answering your questions and providing you with a better understanding of how to solve this equation.
Final Answer
The final answer is x = 9/(4e^6).