Select The Correct Answer.What Is The Solution To This Equation?$\log_4(x^2+1)=\log_4(-2x$\]A. $x=1$B. No SolutionC. $x=-1$D. $x=2$

Feb 27, 2025 by ADMIN 132 views

**Solving Logarithmic Equations: A Step-by-Step Guide**

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore how to solve logarithmic equations, using the equation $\log_4(x^2+1)=\log_4(-2x)$ as an example. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm. The logarithm of a number is the power to which a base number must be raised to produce that number. For example, $\log_4(16)=2$ because $4^2=16$ . Logarithmic equations can be solved using various techniques, including the use of logarithmic properties and the change of base formula.

The Given Equation

The given equation is $\log_4(x^2+1)=\log_4(-2x)$ . This equation involves two logarithms with the same base, which is 4. We can start by using the property of logarithms that states that if $\log_a(x)=\log_a(y)$ , then $x=y$ .

Using the Property of Logarithms

Using the property of logarithms, we can rewrite the given equation as:

$x^2+1=-2x$

This equation is now in a form that we can solve using algebraic techniques.

Solving the Equation

To solve the equation $x^2+1=-2x$ , we can start by moving all the terms to one side of the equation:

$x^2+2x+1=0$

This is a quadratic equation, and we can solve it using the quadratic formula or by factoring.

Factoring the Quadratic Equation

The quadratic equation $x^2+2x+1=0$ can be factored as:

$(x+1)^2=0$

This equation has a repeated root, which means that the solution is $x=-1$ .

Conclusion

In this article, we have solved the logarithmic equation $\log_4(x^2+1)=\log_4(-2x)$ using the property of logarithms and algebraic techniques. We have shown that the solution to the equation is $x=-1$ . This example illustrates the importance of understanding logarithmic properties and how to apply them to solve equations.

Answer

The correct answer is:

C. $x=-1$

Discussion

This equation can be solved using various techniques, including the use of logarithmic properties and the change of base formula. The solution to the equation is $x=-1$ , which can be verified by plugging the value back into the original equation.

Additional Examples

Here are a few more examples of logarithmic equations that can be solved using the same techniques:

$\log_2(x+1)=\log_2(3x)$
$\log_5(x-1)=\log_5(2x)$
$\log_3(x+2)=\log_3(4x)$

These examples illustrate the importance of understanding logarithmic properties and how to apply them to solve equations.

Conclusion

In conclusion, solving logarithmic equations requires a good understanding of logarithmic properties and algebraic techniques. By applying these techniques, we can solve equations involving logarithms and arrive at the correct solution. The examples provided in this article illustrate the importance of understanding logarithmic properties and how to apply them to solve equations.

Final Thoughts

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. By understanding logarithmic properties and applying algebraic techniques, we can solve equations involving logarithms and arrive at the correct solution. The examples provided in this article illustrate the importance of understanding logarithmic properties and how to apply them to solve equations.

References

[1] "Logarithmic Equations" by Math Open Reference
[2] "Solving Logarithmic Equations" by Khan Academy
[3] "Logarithmic Properties" by Wolfram MathWorld

Additional Resources

[1] "Logarithmic Equations" by Mathway
[2] "Solving Logarithmic Equations" by Symbolab
[3] "Logarithmic Properties" by IXL
Logarithmic Equations Q&A ==========================

Frequently Asked Questions

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. The logarithm of a number is the power 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 can use the property of logarithms that states that if $\log_a(x)=\log_a(y)$ , then $x=y$ . You can also use algebraic techniques, such as factoring or the quadratic formula, to solve the equation.

Q: What is the base of a logarithm?

A: The base of a logarithm is the number that is raised to a power to produce the number inside the logarithm. For example, in the equation $\log_4(x^2+1)=\log_4(-2x)$ , the base is 4.

Q: How do I change the base of a logarithm?

A: To change the base of a logarithm, you can use the change of base formula, which is $\log_a(x)=\frac{\log_b(x)}{\log_b(a)}$ . This formula allows you to change the base of a logarithm to any other base.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponential expression. For example, the equation $\log_4(x^2+1)=\log_4(-2x)$ is a logarithmic equation, while the equation $4^x=16$ is an exponential equation.

Q: Can I use a calculator to solve a logarithmic equation?

A: Yes, you can use a calculator to solve a logarithmic equation. However, it's always a good idea to check your work by plugging the solution back into the original equation.

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

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

Not using the property of logarithms that states that if $\log_a(x)=\log_a(y)$ , then $x=y$ .
Not checking your work by plugging the solution back into the original equation.
Not using the change of base formula to change the base of a logarithm.
Not using algebraic techniques, such as factoring or the quadratic formula, to solve the equation.

Q: Can I use logarithmic equations in real-world applications?

A: Yes, logarithmic equations can be used in a variety of real-world applications, including:

Finance: Logarithmic equations can be used to calculate interest rates and investment returns.
Science: Logarithmic equations can be used to model population growth and decay.
Engineering: Logarithmic equations can be used to design and optimize systems.

Q: What are some common types of logarithmic equations?

A: Some common types of logarithmic equations include:

Logarithmic equations with the same base: These are equations that involve two logarithms with the same base, such as $\log_4(x^2+1)=\log_4(-2x)$ .
Logarithmic equations with different bases: These are equations that involve two logarithms with different bases, such as $\log_4(x^2+1)=\log_2(-2x)$ .
Logarithmic equations with multiple logarithms: These are equations that involve multiple logarithms, such as $\log_4(x^2+1)=\log_4(-2x)+\log_4(3x)$ .

Conclusion

Additional Resources

[1] "Logarithmic Equations" by Mathway
[2] "Solving Logarithmic Equations" by Symbolab
[3] "Logarithmic Properties" by IXL

Final Thoughts

Logarithmic equations can be used in a variety of real-world applications, including finance, science, and engineering. By understanding logarithmic properties and applying algebraic techniques, we can solve equations involving logarithms and arrive at the correct solution. The examples provided in this article illustrate the importance of understanding logarithmic properties and how to apply them to solve equations.