What Is The True Solution To The Logarithmic Equation Below?$\[\log _4\left[\log _4(2 X)\right]=1\\]A. \[$x=2\$\]B. \[$x=8\$\]C. \[$x=64\$\]D. \[$x=128\$\]

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will explore a specific logarithmic equation and determine the true solution. The equation in question is $\log _4\left[\log _4(2 x)\right]=1$. We will break down the solution step by step, using the properties of logarithms to simplify the equation and find the value of $x$.

Understanding the Equation

The given equation is $\log _4\left[\log _4(2 x)\right]=1$. To solve this equation, we need to understand the properties of logarithms. The logarithm of a number is the exponent to which a base must be raised to produce that number. In this case, the base is 4, and the logarithm is $\log _4(2 x)$. The equation states that the logarithm of $\log _4(2 x)$ is equal to 1.

Simplifying the Equation

To simplify the equation, we can start by rewriting it in exponential form. Since the logarithm is equal to 1, we can rewrite the equation as $4^1 = \log _4(2 x)$. This simplifies to $4 = \log _4(2 x)$.

Using the Definition of Logarithm

The definition of a logarithm states that if $y = \log _a(x)$, then $a^y = x$. In this case, we have $4 = \log _4(2 x)$. Using the definition of logarithm, we can rewrite this as $4^4 = 2 x$.

Solving for $x$

Now that we have simplified the equation, we can solve for $x$. We have $4^4 = 2 x$, which simplifies to $256 = 2 x$. To solve for $x$, we can divide both sides of the equation by 2, resulting in $x = 128$.

Conclusion

In conclusion, the true solution to the logarithmic equation $\log _4\left[\log _4(2 x)\right]=1$ is $x = 128$. This solution was obtained by simplifying the equation using the properties of logarithms and the definition of logarithm.

Comparison with Other Options

Let's compare our solution with the other options provided:

• Option A: $x = 2$
• Option B: $x = 8$
• Option C: $x = 64$
• Option D: $x = 128$

Our solution, $x = 128$, is the only option that matches the solution we obtained by solving the equation.

Importance of Logarithmic Equations

Logarithmic equations are an essential part of mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we have demonstrated how to solve a logarithmic equation using the properties of logarithms and the definition of logarithm. This skill is crucial in various fields, including physics, engineering, and computer science.

Real-World Applications

Logarithmic equations have numerous real-world applications. For example, they are used in finance to calculate interest rates, in physics to describe the behavior of sound waves, and in computer science to analyze algorithms. In this article, we have demonstrated how to solve a logarithmic equation, which is a fundamental skill required to tackle these real-world applications.

Final Thoughts

In conclusion, the true solution to the logarithmic equation $\log _4\left[\log _4(2 x)\right]=1$ is $x = 128$. This solution was obtained by simplifying the equation using the properties of logarithms and the definition of logarithm. We have also demonstrated the importance of logarithmic equations and their real-world applications. By mastering the skill of solving logarithmic equations, we can tackle a wide range of problems in various fields.

Frequently Asked Questions

• Q: What is the definition of a logarithm? A: The definition of a logarithm states that if $y = \log _a(x)$, then $a^y = x$.
• Q: How do I simplify a logarithmic equation? A: To simplify a logarithmic equation, you can use the properties of logarithms and the definition of logarithm.
• Q: What are the real-world applications of logarithmic equations? A: Logarithmic equations have numerous real-world applications, including finance, physics, and computer science.

References

• [1] "Logarithms" by Khan Academy
• [2] "Logarithmic Equations" by Math Open Reference
• [3] "Real-World Applications of Logarithmic Equations" by Wolfram MathWorld
  

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will provide a comprehensive Q&A section to help you better understand logarithmic equations and how to solve them.

Q&A

Q: What is the definition of a logarithm?

A: The definition of a logarithm states that if $y = \log _a(x)$, then $a^y = x$. This means that the logarithm of a number is the exponent to which a base must be raised to produce that number.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can use the properties of logarithms and the definition of logarithm. Some common properties of logarithms include:

• $\log _a(b) + \log _a(c) = \log _a(bc)$
• $\log _a(b) - \log _a(c) = \log _a(\frac{b}{c})$
• $\log _a(b^c) = c \log _a(b)$

Q: What are the common logarithmic equations?

A: Some common logarithmic equations include:

• $\log _a(x) = y$
• $\log _a(x) + y = z$
• $\log _a(x) - y = z$

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the following steps:

1. Simplify the equation using the properties of logarithms.
2. Use the definition of logarithm to rewrite the equation in exponential form.
3. Solve for the variable.

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

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

• Finance: Logarithmic equations are used to calculate interest rates and returns on investment.
• Physics: Logarithmic equations are used to describe the behavior of sound waves and other physical phenomena.
• Computer Science: Logarithmic equations are used to analyze algorithms and data structures.

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

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

• Forgetting to simplify the equation using the properties of logarithms.
• Not using the definition of logarithm to rewrite the equation in exponential form.
• Not solving for the variable.

Q: How do I check my answer when solving a logarithmic equation?

A: To check your answer when solving a logarithmic equation, you can use the following steps:

1. Plug your answer back into the original equation.
2. Simplify the equation using the properties of logarithms.
3. Check if the equation is true.

Q: What are some common logarithmic equations that are used in real-world applications?

A: Some common logarithmic equations that are used in real-world applications include:

• $\log _a(x) = y$
• $\log _a(x) + y = z$
• $\log _a(x) - y = z$

Q: How do I use logarithmic equations to solve problems in finance?

A: To use logarithmic equations to solve problems in finance, you can use the following steps:

1. Identify the problem and the variables involved.
2. Use logarithmic equations to model the problem.
3. Solve the equation using the properties of logarithms and the definition of logarithm.

Q: How do I use logarithmic equations to solve problems in physics?

A: To use logarithmic equations to solve problems in physics, you can use the following steps:

1. Identify the problem and the variables involved.
2. Use logarithmic equations to model the problem.
3. Solve the equation using the properties of logarithms and the definition of logarithm.

Q: How do I use logarithmic equations to solve problems in computer science?

A: To use logarithmic equations to solve problems in computer science, you can use the following steps:

1. Identify the problem and the variables involved.
2. Use logarithmic equations to model the problem.
3. Solve the equation using the properties of logarithms and the definition of logarithm.

Conclusion

In conclusion, logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. By following the steps outlined in this article, you can better understand logarithmic equations and how to solve them. Remember to simplify the equation using the properties of logarithms, use the definition of logarithm to rewrite the equation in exponential form, and solve for the variable. With practice and patience, you can become proficient in solving logarithmic equations and apply them to real-world problems in finance, physics, and computer science.

Frequently Asked Questions

• Q: What is the definition of a logarithm? A: The definition of a logarithm states that if $y = \log _a(x)$, then $a^y = x$.
• Q: How do I simplify a logarithmic equation? A: To simplify a logarithmic equation, you can use the properties of logarithms and the definition of logarithm.
• Q: What are the real-world applications of logarithmic equations? A: Logarithmic equations have numerous real-world applications, including finance, physics, and computer science.

References

• [1] "Logarithms" by Khan Academy
• [2] "Logarithmic Equations" by Math Open Reference
• [3] "Real-World Applications of Logarithmic Equations" by Wolfram MathWorld