Solve The Equation.${ \ln X + \ln (x-2) = 5 }$ { x \approx \}

$$

Introduction

In this article, we will delve into solving a logarithmic equation involving natural logarithms. The equation $\ln x + \ln (x-2) = 5$ requires us to apply various mathematical techniques to isolate the variable $x$. We will start by understanding the properties of logarithms, then proceed to simplify the equation, and finally, use algebraic manipulations to find the value of $x$.

Understanding Logarithmic Properties

Before we begin solving the equation, it's essential to recall the properties of logarithms. The sum of two logarithms with the same base is equal to the logarithm of the product of their arguments. Mathematically, this can be expressed as:

$$\ln a + \ln b = \ln (ab)$$

Using this property, we can rewrite the given equation as:

$$\ln (x(x-2)) = 5$$

Simplifying the Equation

Now that we have applied the logarithmic property, we can simplify the equation further. The expression inside the logarithm can be expanded as:

$$x(x-2) = x^2 - 2x$$

So, the equation becomes:

$$\ln (x^2 - 2x) = 5$$

Isolating the Variable

To isolate the variable $x$, we need to get rid of the logarithm. We can do this by applying the exponential function to both sides of the equation. Since the base of the logarithm is not specified, we will assume it is the natural logarithm with base $e$. Therefore, we can rewrite the equation as:

$$e^{\ln (x^2 - 2x)} = e^5$$

Using the property of exponential functions, we can simplify the left-hand side of the equation:

$$x^2 - 2x = e^5$$

Solving for $x$

Now that we have isolated the variable $x$, we can solve for its value. We can start by rearranging the equation to form a quadratic equation:

$$x^2 - 2x - e^5 = 0$$

This is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 1$, $b = -2$, and $c = -e^5$. We can solve this equation using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Substituting the values of $a$, $b$, and $c$, we get:

$$x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-e^5)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{2 \pm \sqrt{4 + 4e^5}}{2}$$

Approximating the Value of $x$

To find the approximate value of $x$, we can use a calculator to evaluate the expression under the square root. Plugging in the value of $e^5$, we get:

$$x = \frac{2 \pm \sqrt{4 + 4(148.4131591)}}{2}$$

Simplifying the expression, we get:

$$x = \frac{2 \pm \sqrt{596.6526364}}{2}$$

Evaluating the square root, we get:

$$x = \frac{2 \pm 24.3841113}{2}$$

Therefore, the approximate values of $x$ are:

$$x \approx 13.1920562$$

or

$$x \approx -11.1920562$$

Conclusion

In this article, we have solved the logarithmic equation $\ln x + \ln (x-2) = 5$ using various mathematical techniques. We started by applying the properties of logarithms, then simplified the equation, isolated the variable $x$, and finally, used algebraic manipulations to find the value of $x$. The approximate value of $x$ is $13.1920562$ or $-11.1920562$.

$$

Introduction

In our previous article, we solved the logarithmic equation $\ln x + \ln (x-2) = 5$ using various mathematical techniques. In this article, we will address some of the frequently asked questions related to this equation.

Q: What is the base of the logarithm in the equation?

A: The base of the logarithm in the equation is not specified. However, since the equation involves the natural logarithm, we can assume that the base is $e$.

Q: How do I simplify the equation $\ln (x^2 - 2x) = 5$?

A: To simplify the equation, you can apply the exponential function to both sides of the equation. This will give you $x^2 - 2x = e^5$.

Q: How do I solve the quadratic equation $x^2 - 2x - e^5 = 0$?

A: You can solve the quadratic equation using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Substituting the values of $a$, $b$, and $c$, you get:

$$x = \frac{2 \pm \sqrt{4 + 4e^5}}{2}$$

Q: What are the approximate values of $x$?

A: The approximate values of $x$ are $13.1920562$ and $-11.1920562$.

Q: How do I check the validity of the solutions?

A: To check the validity of the solutions, you can plug the values of $x$ back into the original equation and verify that they satisfy the equation.

Q: What if the equation has no real solutions?

A: If the equation has no real solutions, it means that the quadratic equation $x^2 - 2x - e^5 = 0$ has no real roots. In this case, you can use complex numbers to find the solutions.

Q: Can I use other methods to solve the equation?

A: Yes, you can use other methods to solve the equation, such as graphing or numerical methods. However, the method we used in this article is a straightforward and algebraic approach.

Q: What is the significance of the equation $\ln x + \ln (x-2) = 5$?

A: The equation $\ln x + \ln (x-2) = 5$ is a logarithmic equation that involves the natural logarithm. It is a classic example of a logarithmic equation that can be solved using various mathematical techniques.

Q: Can I apply the same method to other logarithmic equations?

A: Yes, you can apply the same method to other logarithmic equations that involve the natural logarithm. However, you may need to modify the method depending on the specific equation and its properties.

Conclusion

In this article, we have addressed some of the frequently asked questions related to the equation $\ln x + \ln (x-2) = 5$. We have provided step-by-step solutions and explanations to help readers understand the concepts and techniques involved in solving logarithmic equations.