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

Introduction

In the realm of mathematics, solving equations is a fundamental skill, and exponential equations are a significant part of this domain. Exponential equations involve variables in the exponents, and tackling them often requires a strong understanding of logarithmic properties and algebraic manipulation. In this article, we delve into solving a specific exponential equation: 4^(log₄(x+8)) = 4². We will explore the steps involved in finding the value of x that satisfies this equation, providing a clear and comprehensive explanation for math enthusiasts and students alike. This detailed solution will not only help you understand the process but also equip you with the tools to solve similar problems with confidence. Let's embark on this mathematical journey to unravel the mystery behind this equation.

Understanding Exponential Equations and Logarithms

Before diving into the solution, it's crucial to grasp the core concepts of exponential equations and logarithms. Exponential equations are equations where the variable appears in the exponent. They often take the form a^x = b, where 'a' is the base, 'x' is the exponent, and 'b' is the result. Logarithms, on the other hand, are the inverse operation of exponentiation. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. In other words, if a^x = b, then logₐ(b) = x. Understanding this relationship between exponents and logarithms is key to solving exponential equations.

Logarithms come in various forms, but the most common are the common logarithm (base 10) and the natural logarithm (base e). However, in this particular problem, we are dealing with a logarithm with base 4, denoted as log₄. It's essential to remember the properties of logarithms, such as the power rule, the product rule, and the quotient rule, as these will be instrumental in simplifying and solving equations. One particularly useful property for our problem is the identity a^(logₐ(x)) = x, which states that if a base raised to the logarithm of a number with the same base, the result is the number itself. This identity is a direct consequence of the inverse relationship between exponentiation and logarithms.

The interplay between exponential functions and logarithmic functions allows us to transform complex equations into simpler forms that are easier to solve. By applying the appropriate logarithmic properties, we can often isolate the variable and determine its value. This understanding forms the bedrock of our approach to solving the equation at hand.

Deconstructing the Equation: 4^(log₄(x+8)) = 4²

The given equation, 4^(log₄(x+8)) = 4², presents an interesting challenge that requires careful consideration of exponential and logarithmic properties. The left-hand side of the equation features an exponential term with a base of 4, where the exponent is a logarithm with the same base. This is a crucial observation because it allows us to apply the identity a^(logₐ(x)) = x. By recognizing this pattern, we can simplify the equation significantly.

On the right-hand side, we have 4², which is a simple exponential term that can be easily evaluated. Understanding the structure of both sides of the equation is the first step towards finding a solution. The left side combines an exponential function and a logarithmic function, creating a unique opportunity for simplification. The right side provides a clear target value that we aim to achieve by manipulating the left side.

The equation highlights the inverse relationship between exponentiation and logarithms. The logarithm in the exponent essentially undoes the exponentiation, allowing us to isolate the expression inside the logarithm. This interplay between exponential and logarithmic functions is a recurring theme in mathematics, and mastering it is essential for solving a wide range of problems. In this case, the equation is set up in a way that makes it particularly amenable to simplification using the aforementioned logarithmic identity.

Step-by-Step Solution: Unraveling the Value of x

Now, let's dive into the step-by-step solution of the equation 4^(log₄(x+8)) = 4². The first key step is recognizing the logarithmic identity a^(logₐ(x)) = x. Applying this identity to the left-hand side of our equation, we can simplify 4^(log₄(x+8)) to x+8. This simplification significantly reduces the complexity of the equation, transforming it from an exponential equation to a simple linear equation.

With this simplification, our equation now becomes x+8 = 4². The next step is to evaluate the right-hand side. 4² is simply 4 multiplied by itself, which equals 16. So, our equation now reads x+8 = 16. This is a straightforward linear equation that can be solved using basic algebraic techniques. To isolate x, we need to subtract 8 from both sides of the equation.

Subtracting 8 from both sides of x+8 = 16, we get x = 16 - 8, which simplifies to x = 8. Therefore, the solution to the equation 4^(log₄(x+8)) = 4² is x = 8. It's important to verify this solution by substituting it back into the original equation to ensure that it holds true. This step confirms that our algebraic manipulations were correct and that we have indeed found the correct value of x.

Verification: Ensuring the Solution's Accuracy

After finding a potential solution to an equation, it's crucial to verify its accuracy. This step ensures that the solution satisfies the original equation and that no errors were introduced during the solving process. In our case, we found that x = 8 is the solution to the equation 4^(log₄(x+8)) = 4². To verify this, we substitute x = 8 back into the original equation and check if both sides are equal.

Substituting x = 8 into the left-hand side of the equation, we get 4^(log₄(8+8)), which simplifies to 4^(log₄(16)). Now, we need to evaluate log₄(16). Since 4² = 16, log₄(16) equals 2. So, the left-hand side becomes 4², which is 16. The right-hand side of the original equation is also 4², which is 16. Since both sides of the equation are equal when x = 8, our solution is verified.

This verification step not only confirms the accuracy of our solution but also reinforces our understanding of the equation and the properties we used to solve it. It demonstrates that our algebraic manipulations were valid and that the value x = 8 indeed satisfies the original equation. Verification is an essential part of problem-solving in mathematics, providing confidence in the correctness of the answer.

Conclusion

In conclusion, we have successfully solved the exponential equation 4^(log₄(x+8)) = 4² and found that the value of x that satisfies the equation is 8. This solution was achieved by applying the logarithmic identity a^(logₐ(x)) = x, simplifying the equation, and using basic algebraic techniques to isolate x. We also emphasized the importance of verifying the solution by substituting it back into the original equation to ensure its accuracy.

This problem highlights the significance of understanding the relationship between exponential functions and logarithmic functions, as well as the properties that govern them. By mastering these concepts, one can tackle a wide range of exponential and logarithmic equations with confidence. The step-by-step approach we followed in this article provides a clear framework for solving similar problems, emphasizing the importance of simplification, algebraic manipulation, and verification.

Mathematics often presents us with challenges that require careful analysis and application of fundamental principles. By systematically breaking down problems and applying the appropriate tools, we can unravel their complexities and arrive at elegant solutions. This exercise in solving an exponential equation serves as a testament to the power of mathematical reasoning and the satisfaction of finding the correct answer.