What Is The Solution To $4^{\log_4(x+8)} = 4^2$?A. $x = -8$ B. $ X = − 4 X = -4 X = − 4 [/tex] C. $x = 4$ D. $x = 8$

Introduction

In this article, we will delve into the solution of a logarithmic equation involving exponentiation. The equation in question is $4^{\log_4(x+8)} = 4^2$. Our goal is to find the value of $x$ that satisfies this equation. We will break down the solution step by step, using properties of logarithms and exponentiation to simplify the equation and isolate the variable $x$.

Understanding the Equation

The given equation is $4^{\log_4(x+8)} = 4^2$. To begin solving this equation, we need to understand the properties of logarithms and exponentiation. The expression $\log_4(x+8)$ represents the logarithm of $x+8$ to the base $4$. This means that $4^{\log_4(x+8)}$ is equivalent to $x+8$. Similarly, $4^2$ is equal to $16$.

Simplifying the Equation

Using the property of logarithms that states $\log_a(a^x) = x$, we can rewrite the equation as $x+8 = 16$. This simplification is possible because the logarithm and exponentiation operations are inverse operations, and they cancel each other out.

Solving for $x$

Now that we have simplified the equation to $x+8 = 16$, we can solve for $x$ by subtracting $8$ from both sides of the equation. This gives us $x = 16 - 8$, which simplifies to $x = 8$.

Conclusion

In conclusion, the solution to the equation $4^{\log_4(x+8)} = 4^2$ is $x = 8$. This is the value of $x$ that satisfies the given equation. We arrived at this solution by simplifying the equation using properties of logarithms and exponentiation, and then solving for $x$.

Final Answer

The final answer to the equation $4^{\log_4(x+8)} = 4^2$ is $x = 8$. This is the correct solution among the options provided.

Comparison with Options

Let's compare our solution with the options provided:

• Option A: $x = -8$
• Option B: $x = -4$
• Option C: $x = 4$
• Option D: $x = 8$

Our solution, $x = 8$, matches option D. Therefore, the correct answer is option D.

Importance of Logarithmic Equations

Logarithmic equations, like the one we solved in this article, are essential in various fields, including mathematics, physics, and engineering. They are used to model real-world phenomena, such as population growth, chemical reactions, and electrical circuits. Understanding logarithmic equations and how to solve them is crucial for problem-solving and critical thinking.

Tips for Solving Logarithmic Equations

When solving logarithmic equations, it's essential to remember the following tips:

• Use properties of logarithms and exponentiation to simplify the equation.
• Identify the base of the logarithm and use it to rewrite the equation.
• Solve for the variable by isolating it on one side of the equation.
• Check your solution by plugging it back into the original equation.

By following these tips and practicing solving logarithmic equations, you will become proficient in solving these types of equations and be able to apply them to real-world problems.

Common Mistakes to Avoid

When solving logarithmic equations, it's easy to make mistakes. Here are some common mistakes to avoid:

• Failing to simplify the equation using properties of logarithms and exponentiation.
• Not identifying the base of the logarithm and using it to rewrite the equation.
• Not solving for the variable by isolating it on one side of the equation.
• Not checking the solution by plugging it back into the original equation.

By avoiding these common mistakes, you will be able to solve logarithmic equations accurately and efficiently.

Conclusion

In conclusion, solving logarithmic equations requires a deep understanding of properties of logarithms and exponentiation. By simplifying the equation, identifying the base of the logarithm, and solving for the variable, we can arrive at the correct solution. Remember to check your solution by plugging it back into the original equation to ensure accuracy. With practice and patience, you will become proficient in solving logarithmic equations and be able to apply them to real-world problems.

Introduction

Logarithmic equations can be challenging to solve, but with practice and understanding of the underlying concepts, you can become proficient in solving them. In this article, we will address some frequently asked questions (FAQs) about logarithmic equations, providing clear explanations and examples to help you better understand the concepts.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. Logarithmic equations are used to solve problems that involve exponential growth or decay.

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

A: A logarithmic equation involves a logarithm, while an exponential equation involves an exponent. For example, the equation $2^x = 8$ is an exponential equation, while the equation $\log_2(8) = x$ is a logarithmic equation.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can use the properties of logarithms, such as the product rule, quotient rule, and power rule. For example, the equation $\log_2(8) + \log_2(4) = x$ can be simplified using the product rule to $\log_2(32) = x$.

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. Identify the base of the logarithm and use it to rewrite the equation.
3. Solve for the variable by isolating it on one side of the equation.
4. Check your solution by plugging it back into the original equation.

Q: What is the base of a logarithm?

A: The base of a logarithm is the number that is used as the exponent in the logarithmic function. For example, in the equation $\log_2(8) = x$, the base of the logarithm is 2.

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 $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, where $c$ is the new base.

Q: What is the logarithmic identity?

A: The logarithmic identity is $\log_a(a) = 1$, which states that the logarithm of a number to its own base is equal to 1.

Q: How do I use the logarithmic identity to solve an equation?

A: To use the logarithmic identity to solve an equation, you can rewrite the equation using the identity and then solve for the variable.

Q: What is the logarithmic property of addition?

A: The logarithmic property of addition states that $\log_a(b) + \log_a(c) = \log_a(bc)$, which means that the logarithm of the product of two numbers is equal to the sum of their logarithms.

Q: How do I use the logarithmic property of addition to simplify an equation?

A: To use the logarithmic property of addition to simplify an equation, you can rewrite the equation using the property and then simplify the expression.

Q: What is the logarithmic property of subtraction?

A: The logarithmic property of subtraction states that $\log_a(b) - \log_a(c) = \log_a(\frac{b}{c})$, which means that the logarithm of the quotient of two numbers is equal to the difference of their logarithms.

Q: How do I use the logarithmic property of subtraction to simplify an equation?

A: To use the logarithmic property of subtraction to simplify an equation, you can rewrite the equation using the property and then simplify the expression.

Q: What is the logarithmic property of multiplication?

A: The logarithmic property of multiplication states that $\log_a(bc) = \log_a(b) + \log_a(c)$, which means that the logarithm of the product of two numbers is equal to the sum of their logarithms.

Q: How do I use the logarithmic property of multiplication to simplify an equation?

A: To use the logarithmic property of multiplication to simplify an equation, you can rewrite the equation using the property and then simplify the expression.

Q: What is the logarithmic property of division?

A: The logarithmic property of division states that $\log_a(\frac{b}{c}) = \log_a(b) - \log_a(c)$, which means that the logarithm of the quotient of two numbers is equal to the difference of their logarithms.

Q: How do I use the logarithmic property of division to simplify an equation?

A: To use the logarithmic property of division to simplify an equation, you can rewrite the equation using the property and then simplify the expression.

Q: What is the logarithmic property of exponentiation?

A: The logarithmic property of exponentiation states that $\log_a(b^c) = c \log_a(b)$, which means that the logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number.

Q: How do I use the logarithmic property of exponentiation to simplify an equation?

A: To use the logarithmic property of exponentiation to simplify an equation, you can rewrite the equation using the property and then simplify the expression.

Conclusion

In conclusion, logarithmic equations can be challenging to solve, but with practice and understanding of the underlying concepts, you can become proficient in solving them. By using the properties of logarithms, such as the product rule, quotient rule, and power rule, you can simplify and solve logarithmic equations. Remember to check your solution by plugging it back into the original equation to ensure accuracy. With practice and patience, you will become proficient in solving logarithmic equations and be able to apply them to real-world problems.