Solving $4^{\log _4(x+8)}=4^2$ A Step-by-Step Guide

In this article, we will delve into solving the exponential equation $4^{\log _4(x+8)}=4^2$. This type of equation involves logarithms and exponents, requiring a clear understanding of their properties to arrive at the correct solution. We will break down the problem step by step, explaining the reasoning behind each operation and highlighting key concepts. This comprehensive guide will not only help you solve this specific equation but also equip you with the skills to tackle similar problems in the future. Let's embark on this mathematical journey together and unravel the mystery of exponential equations.

Understanding the Fundamentals of Exponential Equations

Before diving into the solution, it's crucial to grasp the fundamental concepts of exponential equations and logarithms. An exponential equation is an equation where the variable appears in the exponent. Logarithms, on the other hand, are the inverse operation of exponentiation. In simpler terms, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Understanding this inverse relationship is key to solving equations involving both exponents and logarithms.

In our given equation, $4^{\log _4(x+8)}=4^2$, we have an exponential expression on the left-hand side and a constant exponential term on the right-hand side. The left-hand side involves a logarithm in the exponent, which is where the magic happens. The property we'll leverage here is the inverse relationship between exponentiation and logarithms: $a^{\log _a(x)} = x$. This property states that if you raise a base a to the power of the logarithm of x to the base a, the result is simply x. This powerful tool allows us to simplify complex expressions and isolate the variable we're trying to solve for.

Furthermore, it's important to remember the basic rules of exponents. When the bases are the same, as in our equation, we can equate the exponents. This principle forms the foundation for our solution strategy. We'll use these concepts and rules as we navigate through the steps to solve for x, making sure to explain each step clearly and concisely.

Step-by-Step Solution: Unraveling the Equation

Now, let's dissect the equation $4^{\log _4(x+8)}=4^2$ step by step. The initial hurdle might seem daunting, but by applying the right principles, we can simplify it effectively.

Step 1: Applying the Inverse Property of Logarithms and Exponents

The most crucial step in solving this equation is recognizing and applying the inverse property of logarithms and exponents. As we discussed earlier, the property $a^{\log _a(x)} = x$ holds the key to simplifying the left-hand side of our equation. In our case, the base a is 4, and the expression inside the logarithm is (x + 8). Applying the property, we get:

$4^{\log _4(x+8)} = x + 8$

This transformation significantly simplifies the equation. The complex exponential expression with a logarithm in the exponent is now reduced to a simple algebraic expression, x + 8. This step highlights the power of understanding and applying fundamental mathematical properties.

Step 2: Equating the Exponents

With the left-hand side simplified, our equation now looks like this:

$x + 8 = 4^2$

The right-hand side of the original equation is $4^2$, which is a constant value. We can calculate this value directly:

$4^2 = 16$

Substituting this value back into the equation, we have:

$x + 8 = 16$

Now, we have a simple linear equation. The exponential part has been successfully dealt with, and we're left with a basic algebraic equation that's easy to solve.

Step 3: Isolating the Variable

To isolate x, we need to get it by itself on one side of the equation. In the equation $x + 8 = 16$, we can do this by subtracting 8 from both sides. This maintains the equality and moves us closer to our solution:

$x + 8 - 8 = 16 - 8$

Simplifying both sides, we get:

$x = 8$

Thus, we have found a potential solution for x. However, it's crucial to verify this solution to ensure it's valid within the context of the original equation.

Verifying the Solution: Ensuring Accuracy

In mathematics, it's always prudent to verify your solution, especially when dealing with logarithms and exponents. This is because logarithms have domain restrictions; the argument of a logarithm (the expression inside the logarithm) must be strictly positive. In our case, the argument of the logarithm is (x + 8), so we need to make sure that (x + 8) is greater than 0 when we substitute our solution for x.

Our potential solution is x = 8. Let's substitute this value back into the argument of the logarithm:

$x + 8 = 8 + 8 = 16$

Since 16 is indeed greater than 0, our solution x = 8 satisfies the domain restriction of the logarithm. This confirms that x = 8 is a valid solution to the original equation.

Substituting x = 8 back into the original equation to fully verify:

$4^{\log _4(8+8)} = 4^{\log _4(16)}$

Since $4^2 = 16$, we have $\log _4(16) = 2$. Therefore:

$4^{\log _4(16)} = 4^2$

This confirms that our solution x = 8 makes the original equation true.

The Final Answer: Unveiling the Solution

After carefully solving the equation and verifying our solution, we arrive at the final answer:

x = 8

This is the only value of x that satisfies the equation $4^{\log _4(x+8)}=4^2$. We have navigated through the complexities of exponential equations and logarithms, applying fundamental properties and principles to arrive at a clear and accurate solution. This process demonstrates the power of mathematical reasoning and the importance of verification.

Common Pitfalls and How to Avoid Them

When tackling exponential and logarithmic equations, there are several common pitfalls that students often encounter. Being aware of these can help you avoid mistakes and arrive at the correct solution more efficiently.

1. Forgetting the Domain Restriction of Logarithms:

As mentioned earlier, the argument of a logarithm must be positive. A common mistake is to solve an equation without considering this restriction, which can lead to extraneous solutions. Always remember to check if your solution makes the argument of the logarithm positive. In our equation, we verified that x = 8 makes (x + 8) positive, ensuring it's a valid solution.

2. Incorrectly Applying Logarithmic Properties:

Logarithms have specific properties that must be applied correctly. For instance, $\log _a(x + y)$ is not the same as $\log _a(x) + \log _a(y)$. Confusing these properties can lead to incorrect simplifications and ultimately, wrong answers. It's crucial to have a firm grasp of logarithmic identities and apply them judiciously.

3. Neglecting to Verify Solutions:

Verification is a crucial step, especially when dealing with equations involving logarithms or radicals. Extraneous solutions can arise due to the nature of these functions. By substituting your solution back into the original equation, you can confirm its validity and avoid incorrect answers. We diligently verified x = 8 in our solution to ensure its accuracy.

4. Misunderstanding the Inverse Relationship:

The inverse relationship between exponents and logarithms is the cornerstone of solving equations like ours. A lack of understanding of this relationship can hinder your ability to simplify expressions effectively. Remember that $a^{\log _a(x)} = x$ and $\log _a(a^x) = x$. Applying this property correctly can drastically simplify the equation.

5. Making Algebraic Errors:

Even if you understand the logarithmic and exponential concepts, simple algebraic errors can derail your solution. Careless mistakes in arithmetic or algebraic manipulations can lead to incorrect answers. Always double-check your steps and ensure accuracy in each operation.

By being mindful of these common pitfalls and practicing consistently, you can develop the skills and confidence to solve exponential and logarithmic equations effectively. Remember, mathematics is a journey of learning and refining your understanding, so embrace the challenges and strive for accuracy.

Practice Problems: Sharpening Your Skills

To solidify your understanding of solving exponential equations involving logarithms, it's essential to practice. Here are a few practice problems that you can try. Remember to apply the principles and steps we've discussed in this article, and always verify your solutions.

1. Solve for x: $5^{\log _5(2x-1)} = 5^3$
2. Solve for x: $2^{\log _2(x^2-3)} = 4$
3. Solve for x: $3^{\log _3(4x+5)} = 9$
4. Solve for x: $10^{\log _{10}(x+2)} = 1000$
5. Solve for x: $e^{\ln(3x-2)} = e^2$

These problems offer a range of scenarios, allowing you to apply your knowledge in different contexts. Work through them methodically, paying attention to each step and verifying your answers. The more you practice, the more comfortable and confident you'll become in solving these types of equations.

Conclusion: Mastering Exponential Equations

In conclusion, solving the exponential equation $4^{\log _4(x+8)}=4^2$ requires a firm grasp of the inverse relationship between exponents and logarithms, as well as attention to detail and verification. By applying the property $a^{\log _a(x)} = x$, we were able to simplify the equation and isolate the variable x. We found that x = 8 is the solution, and we verified this solution to ensure its validity.

Throughout this article, we've emphasized the importance of understanding the underlying principles, avoiding common pitfalls, and practicing consistently. Exponential and logarithmic equations are fundamental concepts in mathematics, and mastering them will open doors to more advanced topics. By following the steps outlined in this guide and tackling practice problems, you can build your skills and confidence in solving these equations.

Remember, mathematics is not just about finding the right answer; it's about understanding the process and the reasoning behind it. Embrace the challenges, ask questions, and never stop learning. With dedication and perseverance, you can conquer any mathematical hurdle and unlock the beauty and power of mathematics.