Solving Logarithmic Equations A Step-by-Step Guide

Hey guys! Today, we're going to dive into solving a logarithmic equation. Logarithmic equations might seem intimidating at first, but trust me, breaking them down into manageable steps makes them a whole lot easier. We'll be tackling the equation $4 \log _6(x+3)=12$. I will guide you through each step, ensuring you not only get the answer but also understand the process. So, let's jump right in and demystify logarithms together!

1. Isolate the Logarithmic Part

Okay, our first mission is to isolate the logarithmic part of the equation. In the equation $4 \log _6(x+3)=12$, the logarithmic part is $\log _6(x+3)$. To isolate it, we need to get rid of that pesky coefficient '4' that's hanging out in front. Think of it like unwrapping a present – we're carefully peeling away the layers to reveal the core. To do this, we'll use a basic algebraic principle: division. Remember, whatever we do to one side of the equation, we have to do to the other to maintain the balance.

So, we'll divide both sides of the equation by 4. This is a crucial step, so let's write it down explicitly:

$\frac{4 \log _6(x+3)}{4} = \frac{12}{4}$

On the left side, the '4's cancel each other out, leaving us with just the logarithmic term. On the right side, 12 divided by 4 is simply 3. This simplifies our equation beautifully to:

$\log _6(x+3) = 3$

Great! We've successfully isolated the logarithmic part. This is a significant milestone because now we can move on to the next step, which involves converting this logarithmic equation into its exponential form. By isolating the logarithm, we've set the stage for solving for 'x' more directly. This step is all about simplifying the equation and making it easier to work with. So, with the logarithm isolated, we're well on our way to finding the solution!

2. Convert to Exponential Form

Now that we've isolated the logarithmic part, it's time to transform our equation into exponential form. This might sound a bit technical, but it's really just about understanding the relationship between logarithms and exponents. Think of them as two sides of the same coin – they're different ways of expressing the same idea.

The logarithmic equation we have is $\log _6(x+3) = 3$. Remember, a logarithm essentially asks the question: "To what power must we raise the base (in this case, 6) to get the argument (in this case, x+3)?" The answer is the value on the right side of the equation, which is 3.

To convert this into exponential form, we use the definition of a logarithm: $\log _b(a) = c$ is equivalent to $b^c = a$. Applying this to our equation, we can rewrite $\log _6(x+3) = 3$ as:

$6^3 = x+3$

See how we've taken the base (6), raised it to the power of the result (3), and set it equal to the argument (x+3)? This is the essence of converting from logarithmic to exponential form. It's like translating from one language to another – the meaning stays the same, but the expression is different.

By converting to exponential form, we've eliminated the logarithm and created a more straightforward equation to solve. Instead of dealing with logarithms, we now have a simple algebraic equation involving an exponent. This makes it much easier to isolate 'x' and find its value. This conversion is a pivotal step in solving logarithmic equations, as it bridges the gap between logarithms and the more familiar world of exponents.

3. Simplify and Solve for x

Alright, we've successfully converted our logarithmic equation into exponential form, and now we're ready to simplify and solve for x. Our equation currently looks like this: $6^3 = x+3$. The first thing we need to do is simplify the exponential term, $6^3$. This means we need to calculate 6 raised to the power of 3. This is where our basic math skills come into play.

$6^3$ is the same as 6 multiplied by itself three times: $6 * 6 * 6$. Let's break it down step by step: 6 * 6 equals 36, and then 36 * 6 equals 216. So, $6^3 = 216$. Now we can substitute this value back into our equation:

$216 = x + 3$

Now, we have a simple linear equation. Our goal is to isolate 'x' on one side of the equation. To do this, we need to get rid of the '+3' that's being added to 'x'. We can achieve this by using the inverse operation – subtraction. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance.

So, we'll subtract 3 from both sides of the equation:

$216 - 3 = x + 3 - 3$

On the left side, 216 minus 3 is 213. On the right side, the '+3' and '-3' cancel each other out, leaving us with just 'x'. This simplifies our equation to:

$213 = x$

We've done it! We've successfully solved for 'x'. The value of 'x' that satisfies the original equation is 213. This step-by-step simplification process is crucial in solving any equation, and by breaking it down into smaller, manageable steps, we've made it much easier to understand and execute.

4. State the Solution

We've reached the final destination – stating the solution! After all the hard work we've put in, it's time to clearly present our answer. We've meticulously solved for x and found that $x = 213$. Now, we need to express this as an equation, just as the instructions requested.

So, our final solution, written as an equation, is:

$x = 213$

It's essential to present your answer clearly and in the format requested, as this demonstrates a thorough understanding of the problem and its requirements. In this case, stating the solution as an equation leaves no room for ambiguity. It's a concise and precise way to communicate our findings.

But hold on a second! Before we declare victory, there's one more crucial step we should always take when dealing with logarithmic equations: checking our solution. This is like the final quality control check to ensure our answer is valid and doesn't lead to any mathematical inconsistencies.

5. Verify the Solution (Important!)

Before we can confidently say we've conquered this logarithmic equation, we need to perform a crucial step: verifying the solution. This is especially important with logarithmic equations because logarithms have domain restrictions. Remember, we can only take the logarithm of a positive number. So, we need to make sure that our solution, $x = 213$, doesn't cause us to take the logarithm of a negative number or zero in the original equation.

Our original equation is $4 \log _6(x+3)=12$. To verify our solution, we'll substitute $x = 213$ back into the original equation and see if it holds true.

Substituting $x = 213$, we get:

$4 \log _6(213+3)=12$

Now, let's simplify inside the logarithm:

$4 \log _6(216)=12$

Next, we need to evaluate $\log _6(216)$. This asks the question: "To what power must we raise 6 to get 216?" We already figured this out earlier when we converted to exponential form – $6^3 = 216$. So, $\log _6(216) = 3$.

Substituting this back into our equation, we get:

$4 * 3 = 12$

Simplifying the left side, we have:

$12 = 12$

This is a true statement! Since our substitution resulted in a true equation, we can confidently say that our solution, $x = 213$, is valid. This verification step is like the final seal of approval, confirming that we've not only found a solution but also that it's mathematically sound.

If we had substituted our solution and ended up with an untrue statement, it would indicate that our solution is extraneous and not a valid answer. In such cases, we would need to re-examine our steps to identify any potential errors.

Conclusion

Woohoo! We've successfully navigated the world of logarithmic equations and solved the equation $4 \log _6(x+3)=12$. We broke down the problem into manageable steps: isolating the logarithmic part, converting to exponential form, simplifying and solving for x, stating the solution, and, most importantly, verifying our answer. We found that $x = 213$ is indeed the solution to the equation.

Remember, solving logarithmic equations is like any other mathematical skill – it gets easier with practice. The key is to understand the underlying principles and follow a systematic approach. So, keep practicing, and you'll become a logarithm-solving pro in no time!

If you ever get stuck, remember these steps, and don't hesitate to revisit the concepts of logarithms and exponents. Happy solving, guys!