What Is The Approximate Solution To This Equation?${ 19 + 2 \ln X = 25 }$A. { X \approx 0.05 $}$ B. { X \approx 20.09 $}$ C. { X \approx 3 $}$ D. { X \approx 1.93 $}$

Introduction

In mathematics, equations are used to represent relationships between variables. Solving equations is a crucial aspect of mathematics, and it involves finding the value of the variable that satisfies the equation. In this article, we will focus on solving a specific equation involving a natural logarithm. We will use algebraic techniques to isolate the variable and find its approximate value.

The Equation

The given equation is:

${ 19 + 2 \ln x = 25 }$

This equation involves a natural logarithm, which is a fundamental concept in mathematics. The natural logarithm is denoted by the symbol $\ln$ and is the inverse of the exponential function. In this equation, the natural logarithm is multiplied by 2, which means that the logarithmic term is amplified.

Step 1: Isolate the Logarithmic Term

To solve this equation, we need to isolate the logarithmic term. We can do this by subtracting 19 from both sides of the equation:

${ 2 \ln x = 25 - 19 }$

Simplifying the right-hand side, we get:

${ 2 \ln x = 6 }$

Step 2: Divide by 2

Next, we need to divide both sides of the equation by 2 to isolate the logarithmic term:

${ \ln x = \frac{6}{2} }$

Simplifying the right-hand side, we get:

${ \ln x = 3 }$

Step 3: Exponentiate Both Sides

To solve for $x$, we need to exponentiate both sides of the equation. Since the natural logarithm is the inverse of the exponential function, we can use the exponential function to "undo" the logarithm:

${ x = e^3 }$

Step 4: Approximate the Value of $x$

To find the approximate value of $x$, we can use a calculator or a numerical method to evaluate the exponential function:

${ x \approx e^3 }$

Using a calculator, we get:

${ x \approx 20.08554 }$

Rounding this value to two decimal places, we get:

${ x \approx 20.09 }$

Conclusion

In this article, we solved the equation $19 + 2 \ln x = 25$ using algebraic techniques. We isolated the logarithmic term, divided by 2, and exponentiated both sides to solve for $x$. The approximate value of $x$ is $20.09$. This solution is consistent with option B.

Comparison of Options

Let's compare our solution with the options provided:

• Option A: $x \approx 0.05$
• Option B: $x \approx 20.09$
• Option C: $x \approx 3$
• Option D: $x \approx 1.93$

Our solution is closest to option B, which is $x \approx 20.09$. This confirms that our solution is correct.

Final Answer

The final answer is:

${ x \approx 20.09 }$

Q: What is the equation $19 + 2 \ln x = 25$ trying to solve?

A: The equation $19 + 2 \ln x = 25$ is trying to solve for the value of $x$ that satisfies the equation. In other words, we need to find the value of $x$ that makes the equation true.

Q: What is the natural logarithm, and how is it used in the equation?

A: The natural logarithm, denoted by $\ln$, is a mathematical function that takes a positive real number as input and returns a real number as output. In the equation $19 + 2 \ln x = 25$, the natural logarithm is used to represent the relationship between the variable $x$ and the constant 19.

Q: Why do we need to isolate the logarithmic term in the equation?

A: We need to isolate the logarithmic term in the equation because it is the only term that contains the variable $x$. By isolating the logarithmic term, we can solve for $x$ and find its value.

Q: What is the difference between the exponential function and the natural logarithm?

A: The exponential function, denoted by $e^x$, is a mathematical function that takes a real number as input and returns a positive real number as output. The natural logarithm, denoted by $\ln$, is the inverse of the exponential function. In other words, the natural logarithm "undoes" the exponential function.

Q: How do we use the exponential function to solve for $x$ in the equation?

A: We use the exponential function to solve for $x$ by exponentiating both sides of the equation. Since the natural logarithm is the inverse of the exponential function, we can use the exponential function to "undo" the logarithm and solve for $x$.

Q: What is the approximate value of $x$ that satisfies the equation?

A: The approximate value of $x$ that satisfies the equation is $x \approx 20.09$. This value is consistent with option B.

Q: Why is it important to use a calculator or numerical method to evaluate the exponential function?

A: It is important to use a calculator or numerical method to evaluate the exponential function because the exponential function can produce very large or very small values. Using a calculator or numerical method allows us to accurately evaluate the exponential function and find the approximate value of $x$.

Q: Can we use other methods to solve for $x$ in the equation?

A: Yes, we can use other methods to solve for $x$ in the equation. For example, we can use numerical methods such as the Newton-Raphson method or the bisection method to find the approximate value of $x$. However, the method we used in this article is a simple and straightforward approach that yields the correct solution.

Q: What is the final answer to the equation?

A: The final answer to the equation is $x \approx 20.09$. This value is consistent with option B.

Conclusion

In this article, we answered frequently asked questions about the equation $19 + 2 \ln x = 25$. We discussed the natural logarithm, the exponential function, and how to use them to solve for $x$ in the equation. We also provided the final answer to the equation, which is $x \approx 20.09$.