Solve The Equation Log ⁡ 2 ( X 2 + 3 X + 8 ) = 3 \log_2\left(x^2 + 3x + 8\right) = 3 Lo G 2 ​ ( X 2 + 3 X + 8 ) = 3 .A. X = − 3 ± I 23 2 X = \frac{-3 \pm I \sqrt{23}}{2} X = 2 − 3 ± I 23 ​ ​ B. X = 3 , 8 X = 3, 8 X = 3 , 8 C. X = − 3 , 0 X = -3, 0 X = − 3 , 0 D. X = 3 , 0 X = 3, 0 X = 3 , 0

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Introduction

In this article, we will delve into solving the equation $\log_2\left(x^2 + 3x + 8\right) = 3$. This equation involves logarithmic functions and quadratic equations, making it a challenging problem to solve. We will break down the solution step by step, using mathematical concepts and techniques to arrive at the final answer.

Step 1: Understanding the Logarithmic Function

The logarithmic function $\log_2(x)$ is the inverse of the exponential function $2^x$. It is defined as the power to which the base 2 must be raised to obtain the number $x$. In this equation, we have $\log_2\left(x^2 + 3x + 8\right) = 3$, which means that $2^3 = x^2 + 3x + 8$.

Step 2: Simplifying the Equation

We can simplify the equation by evaluating $2^3$, which is equal to 8. Therefore, we have $8 = x^2 + 3x + 8$. Subtracting 8 from both sides of the equation, we get $0 = x^2 + 3x$.

Step 3: Factoring the Quadratic Equation

The quadratic equation $x^2 + 3x = 0$ can be factored as $x(x + 3) = 0$. This means that either $x = 0$ or $x + 3 = 0$.

Step 4: Solving for x

Solving the equation $x + 3 = 0$, we get $x = -3$. Therefore, the solutions to the equation are $x = 0$ and $x = -3$.

Conclusion

In conclusion, the solutions to the equation $\log_2\left(x^2 + 3x + 8\right) = 3$ are $x = 0$ and $x = -3$. These solutions can be verified by plugging them back into the original equation.

Final Answer

The final answer is $\boxed{D}$.

Discussion

The equation $\log_2\left(x^2 + 3x + 8\right) = 3$ is a logarithmic equation that involves a quadratic expression. To solve this equation, we used the properties of logarithmic functions and factored the quadratic expression. The solutions to the equation are $x = 0$ and $x = -3$, which can be verified by plugging them back into the original equation.

Related Topics

• Logarithmic equations
• Quadratic equations
• Factoring
• Inverse functions

References

• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Quadratic Equations" by Math Is Fun
• [3] "Factoring" by Purplemath

Additional Resources

• Khan Academy: Logarithmic Equations
• Mathway: Quadratic Equations
• Wolfram Alpha: Factoring

Tags

• logarithmic equations
• quadratic equations
• factoring
• inverse functions
• mathematics
  

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Introduction

In our previous article, we solved the equation $\log_2\left(x^2 + 3x + 8\right) = 3$ and found that the solutions are $x = 0$ and $x = -3$. In this article, we will answer some frequently asked questions about solving this equation.

Q: What is the first step in solving the equation $\log_2\left(x^2 + 3x + 8\right) = 3$?

A: The first step in solving the equation is to understand the logarithmic function and its properties. We need to recall that the logarithmic function $\log_2(x)$ is the inverse of the exponential function $2^x$. This means that $2^3 = x^2 + 3x + 8$.

Q: How do we simplify the equation $2^3 = x^2 + 3x + 8$?

A: We can simplify the equation by evaluating $2^3$, which is equal to 8. Therefore, we have $8 = x^2 + 3x + 8$. Subtracting 8 from both sides of the equation, we get $0 = x^2 + 3x$.

Q: How do we factor the quadratic equation $x^2 + 3x = 0$?

A: The quadratic equation $x^2 + 3x = 0$ can be factored as $x(x + 3) = 0$. This means that either $x = 0$ or $x + 3 = 0$.

Q: What are the solutions to the equation $x + 3 = 0$?

A: Solving the equation $x + 3 = 0$, we get $x = -3$. Therefore, the solutions to the equation are $x = 0$ and $x = -3$.

Q: How do we verify the solutions to the equation?

A: We can verify the solutions by plugging them back into the original equation. Plugging $x = 0$ into the equation, we get $\log_2\left(0^2 + 3(0) + 8\right) = \log_2(8) = 3$. This confirms that $x = 0$ is a solution to the equation. Similarly, plugging $x = -3$ into the equation, we get $\log_2\left((-3)^2 + 3(-3) + 8\right) = \log_2(2) = 1$. This does not confirm that $x = -3$ is a solution to the equation.

Q: What is the final answer to the equation $\log_2\left(x^2 + 3x + 8\right) = 3$?

A: The final answer to the equation is $\boxed{D}$, which corresponds to the solutions $x = 0$ and $x = -3$.

Q: What are some related topics to the equation $\log_2\left(x^2 + 3x + 8\right) = 3$?

A: Some related topics to the equation include logarithmic equations, quadratic equations, factoring, and inverse functions.

Q: Where can I find additional resources to learn more about solving logarithmic equations?

A: You can find additional resources on Khan Academy, Mathway, and Wolfram Alpha.

Tags

• logarithmic equations
• quadratic equations
• factoring
• inverse functions
• mathematics

Related Articles

• Solving the Equation $\log_2\left(x^2 + 3x + 8\right) = 3$
• Logarithmic Equations
• Quadratic Equations
• Factoring
• Inverse Functions

References

• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Quadratic Equations" by Math Is Fun
• [3] "Factoring" by Purplemath

Additional Resources

• Khan Academy: Logarithmic Equations
• Mathway: Quadratic Equations
• Wolfram Alpha: Factoring