Solving Logarithmic Equations Finding Domain And Solutions

Hey guys! Today, we're diving deep into the fascinating world of logarithmic equations. Logarithmic equations might seem intimidating at first, but trust me, with a clear understanding of the core concepts and a step-by-step approach, you'll be solving them like a pro in no time! We're going to break down a specific equation, explore the crucial aspects of domain determination, and then walk through the process of finding the solution. So, let's get started and unlock the secrets of logarithmic equations!

Understanding the Logarithmic Equation

Before we jump into the nitty-gritty, let's take a moment to understand the equation we'll be working with. We have:

$$\log _4(x+2)=\log _4(4 x-5)$$

This equation presents us with a logarithm on both sides, which is excellent news! It means we can leverage some key properties of logarithms to simplify things. The core idea here is that if we have logarithms with the same base on both sides of the equation, we can essentially "cancel them out" and focus on the arguments (the expressions inside the parentheses). However, we need to tread carefully and always remember the domain restrictions that logarithms impose.

Domain Restrictions A Crucial First Step

The domain of a logarithmic function is a critical concept. Remember, logarithms are only defined for positive arguments. You can't take the logarithm of a negative number or zero. This is because logarithms ask the question, "To what power must I raise the base to get this number?" If the number is negative or zero, there's no power that will work. Think about it like this: the base of our logarithm is 4. There's no power you can raise 4 to that will result in a negative number or zero.

Therefore, before we even think about solving the equation, we must determine the domain of the variable x. This involves ensuring that the arguments of both logarithms are strictly greater than zero. We have two arguments in our equation:

1. x + 2
2. 4x - 5

To find the domain, we set up two inequalities:

1. x + 2 > 0
2. 4x - 5 > 0

Solving the first inequality, we subtract 2 from both sides:

x > -2

This tells us that x must be greater than -2.

Now, let's solve the second inequality. We add 5 to both sides:

4x > 5

Then, we divide both sides by 4:

x > 5/4

So, x must also be greater than 5/4. Now, here's the crucial part: we need to consider both of these restrictions. For x, it needs to be greater than both -2 and 5/4. Since 5/4 is greater than -2, the more restrictive condition is x > 5/4. This means our domain for the equation is all values of x greater than 5/4. We can write this in interval notation as (5/4, ∞). This is our safe zone – any solution we find must fall within this range!

Solving the Logarithmic Equation Step-by-Step

Now that we've nailed down the domain, we can confidently move on to solving the equation. Remember, our equation is:

$$\log _4(x+2)=\log _4(4 x-5)$$

The beauty of this equation lies in the fact that we have the same base (4) for the logarithms on both sides. This allows us to apply a fundamental property of logarithms: If log_b(A) = log_b(B), then A = B. In simpler terms, we can equate the arguments of the logarithms.

So, we can rewrite the equation as:

x + 2 = 4x - 5

Now, we have a simple linear equation to solve. Let's get all the x terms on one side and the constants on the other. Subtract x from both sides:

2 = 3x - 5

Add 5 to both sides:

7 = 3x

Finally, divide both sides by 3:

x = 7/3

Great! We've found a potential solution: x = 7/3. But hold on a second! We're not done yet. Remember the domain we carefully determined earlier? We need to check if this solution falls within our safe zone (5/4, ∞).

Verifying the Solution Against the Domain

This is a critical step that many students overlook, and it can lead to incorrect answers. We need to make sure that our solution, x = 7/3, is actually a valid solution by checking if it satisfies the domain restriction x > 5/4.

Let's compare 7/3 and 5/4. To make the comparison easier, we can find a common denominator. The least common multiple of 3 and 4 is 12. So, we can rewrite the fractions as:

7/3 = 28/12 5/4 = 15/12

Now it's clear: 28/12 is indeed greater than 15/12. Therefore, 7/3 > 5/4. Our solution x = 7/3 does fall within the domain (5/4, ∞).

This means x = 7/3 is a valid solution to the logarithmic equation.

Conclusion Mastering Logarithmic Equations

And there you have it! We've successfully navigated the world of logarithmic equations, tackling a problem step-by-step. We started by understanding the equation, then emphasized the crucial importance of determining the domain, solved the equation itself, and finally, verified our solution against the domain. Remember, when dealing with logarithmic equations, always prioritize finding the domain first. This will prevent you from accepting extraneous solutions.

Logarithmic equations can seem tricky at first, but with practice and a solid understanding of the core principles, you'll become a whiz at solving them. Keep practicing, keep exploring, and most importantly, have fun with math! If you guys have any questions, feel free to ask. Until next time, happy solving!