Solving 4^(log₄(x+8)) = 4² A Step By Step Guide

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In this article, we will delve into solving the exponential equation 4^(log₄(x+8)) = 4². This problem combines exponential and logarithmic functions, requiring a solid understanding of their properties to arrive at the correct solution. We will break down the steps involved, offering a detailed explanation for each, ensuring clarity for readers of all backgrounds. This comprehensive guide will not only provide the solution but also enhance your grasp of mathematical concepts, preparing you for similar problems in the future. Exponential equations are fundamental in various fields, including mathematics, physics, and engineering, making this a valuable skill to acquire.

Understanding the Basics

Before we tackle the equation, let's revisit some fundamental concepts. Exponential functions are of the form a^x, where a is the base and x is the exponent. Logarithmic functions, on the other hand, are the inverse of exponential functions. The expression logₐ(b) asks the question: "To what power must we raise a to get b?" A key property to remember is that a^(logₐ(x)) = x. This property is crucial for simplifying equations involving both exponential and logarithmic terms. Additionally, it's important to remember the laws of exponents, such as a^(m) = a^(n) implies m = n if a is a positive number not equal to 1. These basic principles will serve as our foundation for solving the given equation.

Step-by-Step Solution

Now, let's dive into the step-by-step solution of the equation 4^(log₄(x+8)) = 4².

  1. Simplify the left side: The first step involves simplifying the left side of the equation. We can apply the property a^(logₐ(x)) = x. In our case, a = 4 and x = x + 8. Therefore, 4^(log₄(x+8)) simplifies to x + 8. This simplification is a direct application of the fundamental relationship between exponential and logarithmic functions. By recognizing this relationship, we can significantly reduce the complexity of the equation.

  2. Rewrite the equation: After simplification, our equation becomes x + 8 = 4². This step transforms the original exponential equation into a simple linear equation. The simplification process highlights the power of understanding and applying mathematical properties. Now, the equation is in a more manageable form, allowing us to proceed with basic algebraic operations.

  3. Evaluate the right side: Next, we evaluate the right side of the equation, which is 4². 4² is simply 4 multiplied by itself, which equals 16. So, the equation becomes x + 8 = 16. This step is a straightforward arithmetic calculation, but it's crucial for progressing towards the final solution. Accurate evaluation is essential to maintain the integrity of the equation.

  4. Isolate x: To isolate x, we subtract 8 from both sides of the equation. This gives us x + 8 - 8 = 16 - 8, which simplifies to x = 8. This is a standard algebraic manipulation, ensuring that we maintain the equality of the equation. Isolating the variable is a key step in solving any algebraic equation.

  5. Verify the solution: Finally, we need to verify our solution by substituting x = 8 back into the original equation. This step is crucial to ensure that our solution is valid and does not lead to any undefined expressions, especially when dealing with logarithms. Substituting x = 8 into the original equation, we get 4^(log₄(8+8)) = 4^(log₄(16)). Since 4² = 16, log₄(16) = 2. Thus, the left side becomes 4² = 16, which is equal to the right side of the original equation. Therefore, our solution x = 8 is correct.

Detailed Explanation of Each Step

To further clarify the solution process, let's delve into a more detailed explanation of each step.

Step 1: Simplifying the Left Side

The core of this step lies in the property a^(logₐ(x)) = x. This property stems from the inverse relationship between exponential and logarithmic functions. The logarithm logₐ(x) gives the exponent to which a must be raised to obtain x. Therefore, raising a to this exponent should logically result in x. In our equation, we have 4^(log₄(x+8)). Applying the property, we directly replace this expression with x + 8. This simplification is not just a mathematical trick; it's a direct consequence of the definitions of logarithms and exponentials. This step significantly simplifies the equation, making it easier to solve. Understanding this property is essential for handling equations involving both exponential and logarithmic functions.

Step 2: Rewriting the Equation

After simplifying the left side, the equation transforms into x + 8 = 4². This transition marks a significant reduction in complexity. The original equation involved both exponential and logarithmic terms, which can be intimidating. However, after applying the logarithmic property, we are left with a simple linear equation. Linear equations are much easier to solve, involving only basic algebraic manipulations. This step underscores the importance of simplifying equations before attempting to solve them. By reducing the equation to its simplest form, we pave the way for a straightforward solution process. Rewriting the equation in this form makes it accessible and manageable.

Step 3: Evaluating the Right Side

Evaluating the right side of the equation, 4², involves a basic arithmetic calculation. 4² means 4 multiplied by itself, which equals 16. This step is crucial for progressing towards the solution. While it may seem simple, accurate evaluation is paramount. A mistake in this step would propagate through the rest of the solution, leading to an incorrect answer. Therefore, attention to detail is essential even in seemingly straightforward calculations. The equation now becomes x + 8 = 16, a clear and concise form. Accurate evaluation ensures the integrity of the equation and the final solution.

Step 4: Isolating x

Isolating x is a fundamental algebraic technique. To isolate x in the equation x + 8 = 16, we subtract 8 from both sides. This maintains the equality of the equation, a crucial principle in algebra. Subtracting 8 from both sides gives us x + 8 - 8 = 16 - 8, which simplifies to x = 8. This step directly leads us to the solution. The process of isolating the variable is a cornerstone of solving algebraic equations. Isolating x allows us to determine its value and solve the equation.

Step 5: Verifying the Solution

Verifying the solution is a critical step, especially when dealing with logarithmic equations. Logarithms are only defined for positive arguments, so we need to ensure that our solution does not lead to taking the logarithm of a non-positive number. In our case, we substitute x = 8 back into the original equation: 4^(log₄(8+8)) = 4^(log₄(16)). Since 4² = 16, log₄(16) = 2. Thus, the left side becomes 4² = 16, which is equal to the right side of the original equation. This confirms that x = 8 is a valid solution. Verification is essential to ensure the solution's correctness and avoid potential errors.

The Correct Answer

After carefully working through the steps, we have found that the solution to the equation 4^(log₄(x+8)) = 4² is x = 8. Therefore, the correct answer is D. x=8.

Common Mistakes to Avoid

When solving equations like this, it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:

  1. Forgetting the Domain of Logarithms: Logarithms are only defined for positive arguments. This means that the expression inside the logarithm must be greater than zero. In our case, we have log₄(x+8), so x + 8 must be greater than 0. Failing to check this condition can lead to extraneous solutions. Always consider the domain of logarithmic functions when solving equations.

  2. Misapplying Logarithmic Properties: The property a^(logₐ(x)) = x is crucial for simplifying equations like this. However, it's essential to apply it correctly. Make sure you understand the conditions under which this property holds. Misapplying properties can lead to incorrect simplifications and, ultimately, the wrong solution. Proper application of logarithmic properties is key to solving these equations accurately.

  3. Arithmetic Errors: Even simple arithmetic errors can derail the solution process. Be careful when evaluating exponents, adding, subtracting, multiplying, and dividing. Double-check your calculations to ensure accuracy. Small errors can have a significant impact on the final result. Attention to detail in arithmetic calculations is vital.

  4. Not Verifying the Solution: As we emphasized earlier, verifying the solution is a crucial step. It helps catch any mistakes made during the solution process and ensures that the solution is valid. Skipping this step can lead to accepting incorrect solutions. Always verify your solution by substituting it back into the original equation.

Conclusion

In conclusion, solving the equation 4^(log₄(x+8)) = 4² involves applying the properties of exponential and logarithmic functions, performing algebraic manipulations, and verifying the solution. By understanding the steps involved and avoiding common mistakes, you can confidently tackle similar problems. Remember, mathematics is a journey of understanding, and each problem solved adds to your knowledge and skills. Keep practicing, and you'll become more proficient in solving complex equations. Mastering exponential and logarithmic equations is a valuable skill in mathematics and related fields.