Solve The Equation: $|2|^{4x} = 11$

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Introduction

In this article, we will delve into solving the equation $|2|^{4x} = 11$. This equation involves absolute value and exponentiation, making it a bit more complex than a standard linear equation. We will break down the solution step by step, using mathematical concepts and techniques to arrive at the final answer.

Understanding Absolute Value and Exponentiation

Before we dive into solving the equation, let's take a moment to understand the concepts of absolute value and exponentiation.

Absolute Value: The absolute value of a number is its distance from zero on the number line. It is denoted by the symbol $|x|$ and is always non-negative. For example, $|3| = 3$ and $|-3| = 3$.

Exponentiation: Exponentiation is a mathematical operation that involves raising a number to a power. It is denoted by the symbol $a^b$, where $a$ is the base and $b$ is the exponent. For example, $2^3 = 8$.

Solving the Equation

Now that we have a basic understanding of absolute value and exponentiation, let's move on to solving the equation $|2|^{4x} = 11$.

Since the absolute value of 2 is 2, we can rewrite the equation as $2^{4x} = 11$.

Next, we can take the logarithm of both sides of the equation to get rid of the exponent. We will use the natural logarithm, denoted by $\ln$, which is the inverse of the exponential function.

$\ln(2^{4x}) = \ln(11)$

Using the property of logarithms that states $\ln(a^b) = b\ln(a)$, we can rewrite the equation as:

$4x\ln(2) = \ln(11)$

Now, we can divide both sides of the equation by $4\ln(2)$ to solve for $x$:

$x = \frac{\ln(11)}{4\ln(2)}$

Evaluating the Solution

Now that we have the solution to the equation, let's evaluate it to see if it makes sense.

We can use a calculator to evaluate the expression $\frac{\ln(11)}{4\ln(2)}$. Plugging in the values, we get:

$x \approx \frac{2.397895}{4 \times 0.693147} \approx \frac{2.397895}{2.772588} \approx 0.863$

So, the solution to the equation is approximately $x \approx 0.863$.

Conclusion

In this article, we solved the equation $|2|^{4x} = 11$ using mathematical concepts and techniques. We started by understanding the concepts of absolute value and exponentiation, and then moved on to solving the equation using logarithms. Finally, we evaluated the solution to see if it made sense.

Additional Tips and Tricks

Here are some additional tips and tricks that may be helpful when solving equations involving absolute value and exponentiation:

• Use the properties of logarithms: Logarithms can be a powerful tool when solving equations involving exponentiation. Remember that $\ln(a^b) = b\ln(a)$.
• Simplify the equation: Before solving the equation, try to simplify it by canceling out any common factors.
• Check your solution: Once you have a solution, be sure to check it to make sure it makes sense.

Frequently Asked Questions

Here are some frequently asked questions that may be helpful when solving equations involving absolute value and exponentiation:

• What is the difference between absolute value and exponentiation?
  • Absolute value is the distance of a number from zero on the number line, while exponentiation is a mathematical operation that involves raising a number to a power.
• How do I solve an equation involving absolute value and exponentiation?
  • To solve an equation involving absolute value and exponentiation, start by simplifying the equation and then use logarithms to get rid of the exponent.
• What is the solution to the equation $|2|^{4x} = 11$?
  • The solution to the equation $|2|^{4x} = 11$ is approximately $x \approx 0.863$.

Final Thoughts

Solving equations involving absolute value and exponentiation can be a bit challenging, but with the right techniques and tools, it is definitely possible. Remember to use the properties of logarithms, simplify the equation, and check your solution to make sure it makes sense. With practice and patience, you will become a master of solving equations involving absolute value and exponentiation.

Introduction

In our previous article, we solved the equation $|2|^{4x} = 11$ using mathematical concepts and techniques. In this article, we will answer some frequently asked questions about solving equations involving absolute value and exponentiation.

Q&A

Q: What is the difference between absolute value and exponentiation?

A: Absolute value is the distance of a number from zero on the number line, while exponentiation is a mathematical operation that involves raising a number to a power.

Q: How do I solve an equation involving absolute value and exponentiation?

A: To solve an equation involving absolute value and exponentiation, start by simplifying the equation and then use logarithms to get rid of the exponent. For example, if you have the equation $|2|^{4x} = 11$, you can rewrite it as $2^{4x} = 11$ and then take the logarithm of both sides to get $4x\ln(2) = \ln(11)$.

Q: What is the solution to the equation $|2|^{4x} = 11$?

A: The solution to the equation $|2|^{4x} = 11$ is approximately $x \approx 0.863$.

Q: How do I simplify an equation involving absolute value and exponentiation?

A: To simplify an equation involving absolute value and exponentiation, look for any common factors that can be canceled out. For example, if you have the equation $|2x|^{4x} = 11$, you can rewrite it as $2x^{4x} = 11$ and then cancel out the common factor of $2$.

Q: What are some common mistakes to avoid when solving equations involving absolute value and exponentiation?

A: Some common mistakes to avoid when solving equations involving absolute value and exponentiation include:

• Not simplifying the equation: Make sure to simplify the equation before solving it.
• Not using logarithms: Logarithms can be a powerful tool when solving equations involving exponentiation.
• Not checking the solution: Make sure to check the solution to make sure it makes sense.

Q: How do I check my solution to an equation involving absolute value and exponentiation?

A: To check your solution to an equation involving absolute value and exponentiation, plug the solution back into the original equation and make sure it is true. For example, if you have the solution $x \approx 0.863$ to the equation $|2|^{4x} = 11$, you can plug it back into the original equation to get $|2|^{4(0.863)} \approx 11$.

Q: What are some real-world applications of solving equations involving absolute value and exponentiation?

A: Solving equations involving absolute value and exponentiation has many real-world applications, including:

• Physics: Solving equations involving absolute value and exponentiation is used to model real-world phenomena such as motion and energy.
• Engineering: Solving equations involving absolute value and exponentiation is used to design and optimize systems such as electrical circuits and mechanical systems.
• Computer Science: Solving equations involving absolute value and exponentiation is used to develop algorithms and data structures for solving complex problems.

Conclusion

Solving equations involving absolute value and exponentiation can be a bit challenging, but with the right techniques and tools, it is definitely possible. Remember to use the properties of logarithms, simplify the equation, and check your solution to make sure it makes sense. With practice and patience, you will become a master of solving equations involving absolute value and exponentiation.

Additional Resources

Here are some additional resources that may be helpful when solving equations involving absolute value and exponentiation:

• Textbooks: There are many textbooks available that cover the topic of solving equations involving absolute value and exponentiation.
• Online Resources: There are many online resources available that provide tutorials and examples of solving equations involving absolute value and exponentiation.
• Practice Problems: Practice problems are a great way to reinforce your understanding of solving equations involving absolute value and exponentiation.

Final Thoughts

Solving equations involving absolute value and exponentiation is an important skill that has many real-world applications. With practice and patience, you will become a master of solving these types of equations. Remember to use the properties of logarithms, simplify the equation, and check your solution to make sure it makes sense.