Expand: Ln ⁡ ( X 2 − X − 2 ) = \ln \left(x^2-x-2\right) = Ln ( X 2 − X − 2 ) =

$$

=====================================================

Introduction

---

The natural logarithm function, denoted by $\ln$, is a fundamental concept in mathematics, particularly in calculus and analysis. It is used to find the inverse of the exponential function, which is essential in solving various mathematical problems. In this article, we will focus on expanding the natural logarithm of a quadratic expression, specifically $\ln \left(x^2-x-2\right)$.

Understanding the Natural Logarithm

---

The natural logarithm of a number $x$ is defined as the power to which the base $e$ must be raised to produce the number $x$. In other words, $\ln x$ is the exponent to which $e$ must be raised to obtain $x$. This can be expressed mathematically as:

$$\ln x = \log_e x$$

where $\log_e x$ is the logarithm of $x$ to the base $e$.

Expanding the Natural Logarithm

---

To expand the natural logarithm of a quadratic expression, we can use the properties of logarithms. Specifically, we can use the fact that the logarithm of a product is equal to the sum of the logarithms of the factors. This can be expressed mathematically as:

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

Using this property, we can expand the natural logarithm of the quadratic expression $x^2-x-2$ as follows:

$$\ln \left(x^2-x-2\right) = \ln \left(x^2\right) - \ln \left(x+2\right)$$

Simplifying the Expression

---

To simplify the expression further, we can use the fact that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. This can be expressed mathematically as:

$$\ln a^b = b \ln a$$

Using this property, we can simplify the expression as follows:

$$\ln \left(x^2\right) - \ln \left(x+2\right) = 2 \ln x - \ln \left(x+2\right)$$

Using the Quadratic Formula

---

To further simplify the expression, we can use the quadratic formula to factorize the quadratic expression $x^2-x-2$. The quadratic formula is given by:

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

In this case, $a=1$, $b=-1$, and $c=-2$. Plugging these values into the quadratic formula, we get:

$$x = \frac{1 \pm \sqrt{1+8}}{2} = \frac{1 \pm \sqrt{9}}{2} = \frac{1 \pm 3}{2}$$

This gives us two possible values for $x$, namely $x=2$ and $x=-1$.

Substituting the Values of $x$

---

Substituting the values of $x$ into the expression $2 \ln x - \ln \left(x+2\right)$, we get:

$$2 \ln 2 - \ln \left(2+2\right) = 2 \ln 2 - \ln 4$$

$$2 \ln (-1) - \ln \left(-1+2\right) = 2 \ln (-1) - \ln 1$$

Evaluating the Expression

---

Evaluating the expression $2 \ln 2 - \ln 4$, we get:

$$2 \ln 2 - \ln 4 = 2 \ln 2 - 2 \ln 2 = 0$$

Evaluating the expression $2 \ln (-1) - \ln 1$, we get:

$$2 \ln (-1) - \ln 1 = 2 \ln (-1) - 0 = 2 \ln (-1)$$

Conclusion

---

In conclusion, we have expanded the natural logarithm of a quadratic expression, specifically $\ln \left(x^2-x-2\right)$. We have used the properties of logarithms to simplify the expression and have evaluated the expression for two possible values of $x$. The final answer is $\boxed{0}$ for $x=2$ and $\boxed{2 \ln (-1)}$ for $x=-1$.

Final Answer

---

The final answer is $\boxed{0}$ for $x=2$ and $\boxed{2 \ln (-1)}$ for $x=-1$.

References

---

• [1] "Logarithm" by Wikipedia. Retrieved 2023-02-20.
• [2] "Quadratic Formula" by Math Open Reference. Retrieved 2023-02-20.
• [3] "Natural Logarithm" by Wolfram MathWorld. Retrieved 2023-02-20.
  

=====================================================

Introduction

---

In our previous article, we expanded the natural logarithm of a quadratic expression, specifically $\ln \left(x^2-x-2\right)$. We used the properties of logarithms to simplify the expression and evaluated the expression for two possible values of $x$. In this article, we will answer some frequently asked questions related to expanding the natural logarithm of a quadratic expression.

Q&A

---

Q: What is the natural logarithm of a quadratic expression?

A: The natural logarithm of a quadratic expression is the logarithm of the expression with base $e$. It can be expressed mathematically as $\ln \left(ax^2+bx+c\right)$, where $a$, $b$, and $c$ are constants.

Q: How do I expand the natural logarithm of a quadratic expression?

A: To expand the natural logarithm of a quadratic expression, you can use the properties of logarithms. Specifically, you can use the fact that the logarithm of a product is equal to the sum of the logarithms of the factors. This can be expressed mathematically as:

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

Q: What are the properties of logarithms that I can use to expand the natural logarithm of a quadratic expression?

A: There are several properties of logarithms that you can use to expand the natural logarithm of a quadratic expression. These include:

• The logarithm of a product is equal to the sum of the logarithms of the factors: $\ln (ab) = \ln a + \ln b$
• The logarithm of a power is equal to the exponent multiplied by the logarithm of the base: $\ln a^b = b \ln a$
• The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor: $\ln \left(\frac{a}{b}\right) = \ln a - \ln b$

Q: How do I simplify the expression after expanding the natural logarithm of a quadratic expression?

A: To simplify the expression after expanding the natural logarithm of a quadratic expression, you can use the properties of logarithms. Specifically, you can use the fact that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. This can be expressed mathematically as:

$$\ln a^b = b \ln a$$

Q: What are some common mistakes to avoid when expanding the natural logarithm of a quadratic expression?

A: Some common mistakes to avoid when expanding the natural logarithm of a quadratic expression include:

• Not using the properties of logarithms correctly
• Not simplifying the expression after expanding the natural logarithm of a quadratic expression
• Not evaluating the expression for the correct values of $x$

Q: How do I evaluate the expression after expanding the natural logarithm of a quadratic expression?

A: To evaluate the expression after expanding the natural logarithm of a quadratic expression, you need to substitute the values of $x$ into the expression. You can use the quadratic formula to find the values of $x$ that satisfy the quadratic equation.

Q: What are some real-world applications of expanding the natural logarithm of a quadratic expression?

A: Expanding the natural logarithm of a quadratic expression has several real-world applications, including:

• Modeling population growth and decline
• Analyzing financial data
• Solving optimization problems

Conclusion

---

In conclusion, expanding the natural logarithm of a quadratic expression is a fundamental concept in mathematics that has several real-world applications. By understanding the properties of logarithms and how to simplify the expression after expanding the natural logarithm of a quadratic expression, you can solve a wide range of mathematical problems.

Final Answer

---

The final answer is $\boxed{0}$ for $x=2$ and $\boxed{2 \ln (-1)}$ for $x=-1$.

References

---

• [1] "Logarithm" by Wikipedia. Retrieved 2023-02-20.
• [2] "Quadratic Formula" by Math Open Reference. Retrieved 2023-02-20.
• [3] "Natural Logarithm" by Wolfram MathWorld. Retrieved 2023-02-20.