Solving Logarithmic Equations A Step-by-Step Guide For Log Base 4 Of (x+2) = 3

Hey guys! Let's dive into the exciting world of logarithms and tackle the equation $\log _4(x+2)=3$. If you've ever felt a little lost when faced with these types of problems, don't worry! We're going to break it down step by step, making sure you understand the logic behind each move. By the end of this guide, you'll not only know how to solve this specific equation but also have a solid foundation for tackling other logarithmic challenges. So, buckle up and get ready to master those logs!

Understanding Logarithms: The Basics

Before we jump into solving the equation, let's quickly refresh our understanding of what a logarithm actually is. Think of a logarithm as the inverse operation of exponentiation. If you have an exponential expression like $b^y = x$, the logarithm asks the question: "To what power must I raise the base 'b' to get 'x'?" The answer to that question is 'y', and we write it as $\log_b(x) = y$. So, in essence, a logarithm helps us find the exponent. To really nail this down, let's walk through some key components. The base of the logarithm (in our example, 'b') is the number that's being raised to a power. The argument of the logarithm (in our example, 'x') is the value we're trying to obtain by raising the base to a certain power. And the logarithm itself (in our example, 'y') is the exponent we need. This relationship is super important for solving logarithmic equations. Recognizing how logarithms and exponents relate to each other is the golden ticket to simplifying and ultimately solving these problems. Remember, $\log_b(x) = y$ is just another way of saying $b^y = x$. We'll use this back-and-forth throughout our solution. So, before moving on, make sure you're comfortable switching between these two forms – it'll make everything much smoother!

Converting Logarithmic Form to Exponential Form

Now, let's apply this understanding to our specific equation: $\log _4(x+2)=3$. The first crucial step in solving this equation is to convert it from logarithmic form into its equivalent exponential form. Remember that awesome relationship we just discussed? We're going to use it right now. In our equation, the base of the logarithm is 4, the argument is (x+2), and the logarithm itself is 3. Based on the definition $\log_b(x) = y$ being equivalent to $b^y = x$, we can rewrite our equation as $4^3 = x + 2$. See how we took the base (4), raised it to the power of the logarithm (3), and set it equal to the argument (x+2)? This conversion is the key that unlocks the rest of the solution. It transforms a seemingly complex logarithmic equation into a much simpler algebraic one that we can easily handle. Guys, mastering this conversion is a fundamental skill in solving logarithmic equations. It's like having a superpower that lets you rewrite the problem in a way that makes it much more approachable. Take a moment to appreciate how powerful this step is. By simply changing the form of the equation, we've set ourselves up for success. Now, with our equation in exponential form, the path to finding 'x' becomes much clearer. We've essentially removed the "log" mystery and replaced it with a familiar algebraic structure. So, let's keep going and solve for 'x'!

Simplifying and Solving for x

Alright, now that we've transformed our logarithmic equation into exponential form, we have $4^3 = x + 2$. Let's simplify this! First, we need to calculate $4^3$, which means 4 multiplied by itself three times: 4 * 4 * 4 = 64. So, our equation now looks like this: 64 = x + 2. See how much simpler it's become? Now, we're dealing with a basic algebraic equation. Our goal is to isolate 'x' on one side of the equation. To do this, we need to get rid of the '+ 2' on the right side. The inverse operation of addition is subtraction, so we'll subtract 2 from both sides of the equation. This gives us: 64 - 2 = x + 2 - 2. Simplifying both sides, we get 62 = x. Boom! We've solved for 'x'. The value of 'x' that satisfies the original logarithmic equation is 62. This process of simplifying and isolating the variable is a core technique in algebra, and it's essential for solving equations of all kinds. By performing the same operation on both sides of the equation, we maintain the equality while bringing us closer to our solution. In this case, subtracting 2 from both sides was the magic move that unveiled the value of 'x'. Remember, guys, each step we take is about making the equation simpler and more transparent. By carefully applying algebraic principles, we can unravel even the most complex-looking problems. So, with x = 62 in hand, let's do one more thing to make sure our answer is rock solid.

Verifying the Solution

Before we celebrate our victory, there's one crucial step we must take: verifying our solution. It's like the final checkmark on a job well done, ensuring that our answer is not only mathematically correct but also makes sense in the context of the original equation. Remember, logarithmic functions have certain restrictions, particularly regarding their domain. The argument of a logarithm (the part inside the parentheses) must always be positive. So, in our original equation, $\log _4(x+2)=3$, the expression (x+2) must be greater than zero. This is a crucial point! If our solution for 'x' makes (x+2) negative or zero, it's not a valid solution. It's an extraneous solution. To verify, we'll plug our solution, x = 62, back into the original equation and see if it holds true. So, we have $\log _4(62+2)=3$. Simplifying the inside of the parentheses, we get $\log _4(64)=3$. Now, we ask ourselves: "To what power must we raise 4 to get 64?" We know that $4^3 = 64$, so $\log _4(64)$ indeed equals 3. Our equation holds true! And, even more importantly, 62 + 2 = 64, which is positive, so we're in the clear regarding the domain restriction. This verification step gives us confidence that our solution is correct and valid. It's a vital habit to develop when solving any mathematical problem, especially those involving logarithms or other functions with domain restrictions. So, always remember to verify – it's the ultimate peace of mind!

Conclusion

Awesome job, guys! We've successfully solved the logarithmic equation $\log _4(x+2)=3$ and verified our solution. We started by understanding the fundamentals of logarithms and how they relate to exponents. Then, we converted the equation from logarithmic form to exponential form, which was the key to simplifying the problem. After that, we used basic algebraic techniques to isolate 'x' and find its value. And finally, we verified our solution to ensure its validity. This step-by-step approach is a powerful tool for tackling any logarithmic equation. By breaking down the problem into manageable steps and understanding the underlying principles, you can confidently conquer even the trickiest challenges. Remember, the key is to practice and build your understanding. The more you work with logarithms, the more comfortable and confident you'll become. So, keep exploring, keep learning, and keep solving! You've got this!