Solve For X X X If Log ⁡ 2 ( X 2 − 3 X + 10 ) = 3 \log_2(x^2 - 3x + 10) = 3 Lo G 2 ​ ( X 2 − 3 X + 10 ) = 3 .

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Introduction to Logarithmic Equations

Logarithmic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science. In this article, we will focus on solving a logarithmic equation involving a quadratic expression. The given equation is $\log_2(x^2 - 3x + 10) = 3$. Our goal is to find the value of $x$ that satisfies this equation.

Understanding the Properties of Logarithms

Before we dive into solving the equation, it's essential to understand the properties of logarithms. The logarithm of a number is the exponent to which a base number must be raised to produce that number. In this case, we have a base of 2, and the logarithm is equal to 3. This means that $2^3 = x^2 - 3x + 10$. We can rewrite this equation as $8 = x^2 - 3x + 10$.

Solving the Quadratic Equation

Now that we have a quadratic equation, we can use various methods to solve for $x$. One common method is to use the quadratic formula, which states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In our case, the equation is $x^2 - 3x + 2 = 0$, and we can use the quadratic formula to find the solutions.

Applying the Quadratic Formula

Using the quadratic formula, we get:

$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(2)}}{2(1)}$ $x = \frac{3 \pm \sqrt{9 - 8}}{2}$ $x = \frac{3 \pm \sqrt{1}}{2}$

Simplifying the Solutions

Simplifying the solutions, we get:

$x = \frac{3 + 1}{2}$ $x = \frac{3 - 1}{2}$ $x = 2$ $x = 1$

Checking the Solutions

Now that we have the solutions, we need to check if they satisfy the original equation. We can plug in each solution into the original equation to see if it's true.

Checking the Solution $x = 2$

Plugging in $x = 2$ into the original equation, we get:

$\log_2(2^2 - 3(2) + 10) = 3$ $\log_2(4 - 6 + 10) = 3$ $\log_2(8) = 3$ $2^3 = 8$ $8 = 8$

Checking the Solution $x = 1$

Plugging in $x = 1$ into the original equation, we get:

$\log_2(1^2 - 3(1) + 10) = 3$ $\log_2(1 - 3 + 10) = 3$ $\log_2(8) = 3$ $2^3 = 8$ $8 = 8$

Conclusion

In this article, we solved a logarithmic equation involving a quadratic expression. We used the properties of logarithms to rewrite the equation and then solved the resulting quadratic equation using the quadratic formula. We found two solutions, $x = 2$ and $x = 1$, and checked if they satisfy the original equation. Both solutions were found to be true, and therefore, the final answer is $x = 2$ and $x = 1$.

Final Answer

The final answer is $\boxed{2, 1}$.

Additional Tips and Tricks

• When solving logarithmic equations, it's essential to understand the properties of logarithms and how to rewrite the equation.
• The quadratic formula is a powerful tool for solving quadratic equations, but it's not the only method.
• When checking the solutions, make sure to plug in each solution into the original equation to ensure that it's true.

Related Topics

• Logarithmic equations
• Quadratic equations
• Quadratic formula
• Properties of logarithms

References

• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Quadratic Equations" by Math Is Fun
• [3] "Quadratic Formula" by Khan Academy
  

Introduction

Logarithmic equations can be a challenging topic for many students. In our previous article, we solved a logarithmic equation involving a quadratic expression. In this article, we will answer some frequently asked questions about logarithmic equations.

Q&A

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. The logarithm of a number is the exponent to which a base number must be raised to produce that number.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to understand the properties of logarithms and how to rewrite the equation. You can use various methods such as the quadratic formula or factoring to solve the resulting equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I check if a solution is true?

A: To check if a solution is true, you need to plug it back into the original equation. If the equation is true, then the solution is valid.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

• Not understanding the properties of logarithms
• Not rewriting the equation correctly
• Not checking the solutions
• Not using the correct method to solve the equation

Q: Can you give an example of a logarithmic equation?

A: Yes, here is an example of a logarithmic equation:

$\log_2(x^2 - 3x + 10) = 3$

This equation can be solved using the quadratic formula or factoring.

Q: How do I know if a logarithmic equation has a solution?

A: To determine if a logarithmic equation has a solution, you need to check if the equation is true for any value of x. If the equation is true for any value of x, then it has a solution.

Q: What are some real-world applications of logarithmic equations?

A: Logarithmic equations have many real-world applications, including:

• Physics: Logarithmic equations are used to describe the motion of objects under the influence of gravity.
• Engineering: Logarithmic equations are used to design and optimize systems.
• Computer Science: Logarithmic equations are used in algorithms and data structures.

Conclusion

In this article, we answered some frequently asked questions about logarithmic equations. We covered topics such as the definition of a logarithmic equation, how to solve a logarithmic equation, and common mistakes to avoid. We also provided some real-world applications of logarithmic equations.

Final Tips and Tricks

• Make sure to understand the properties of logarithms before solving a logarithmic equation.
• Use the correct method to solve the equation, such as the quadratic formula or factoring.
• Check the solutions to ensure that they are true.
• Practice solving logarithmic equations to become more comfortable with the topic.

Related Topics

• Logarithmic equations
• Quadratic equations
• Quadratic formula
• Properties of logarithms

References

• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Quadratic Equations" by Math Is Fun
• [3] "Quadratic Formula" by Khan Academy