Solving 3 Log₄ X = Log₄ 32 + Log₄ 2 A Step-by-Step Guide
In this article, we will delve into the process of solving a logarithmic equation. Logarithmic equations might seem daunting at first, but with a clear understanding of logarithmic properties and algebraic manipulation, they can be tackled with confidence. We will break down the solution step by step, ensuring that each stage is clearly explained and understood.
The equation we aim to solve is:
3 log₄ x = log₄ 32 + log₄ 2
This equation combines several key aspects of logarithmic functions that are crucial to comprehend before beginning to find a solution. We will utilize logarithmic properties, such as the power rule and the product rule, to simplify the equation. We will also employ algebraic manipulations to isolate the variable and, therefore, find its value.
The study of logarithms is a fascinating and important branch of mathematics. Logarithmic functions are closely related to exponential functions, and they are widely used in science, engineering, and other fields. Logarithms can help simplify complex calculations, model natural phenomena, and offer valuable insights into how different quantities relate to one another.
As we work through this equation, we will emphasize the significance of accuracy and attention to detail. Every step is critical, and a small mistake can lead to an incorrect answer. We will also emphasize the value of reviewing and verifying your solution to confirm that it satisfies the original equation. This technique will not only provide you confidence in your response but will also strengthen your problem-solving abilities.
Whether you're a student learning about logarithms for the first time or someone seeking to refresh your skills, this comprehensive guide will provide you with the tools and understanding needed to solve logarithmic equations successfully. Let’s begin our exploration and unravel the solution to the equation provided.
Understanding Logarithmic Properties
Before we dive into solving the logarithmic equation, it's crucial to understand the fundamental logarithmic properties. These properties are the building blocks for simplifying and solving logarithmic equations. Mastering these properties will make the process much smoother and ensure that you approach each problem with a solid foundation.
The main logarithmic properties we will use in solving the equation 3 log₄ x = log₄ 32 + log₄ 2 are the power rule and the product rule. Let's explore each of these in more detail:
1. Power Rule of Logarithms
The power rule of logarithms states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number. Mathematically, this can be expressed as:
logₐ (bⁿ) = n logₐ b
In simpler terms, if you have an exponent inside a logarithm, you can bring that exponent down and multiply it by the logarithm. This rule is incredibly useful for simplifying equations where the variable or constant is raised to a power within a logarithmic function.
For example, consider the expression log₂ (8³). Using the power rule, we can rewrite this as 3 log₂ 8. Since 8 is 2³, we know that log₂ 8 equals 3. Therefore, the expression simplifies to 3 * 3 = 9.
In our equation, 3 log₄ x = log₄ 32 + log₄ 2, the power rule will be applied to the term 3 log₄ x. By understanding and correctly applying this rule, we can transform this term into log₄ (x³), which is a crucial step in simplifying the equation and moving towards the solution.
The power rule not only simplifies expressions but also makes equations easier to solve. It helps in manipulating the logarithmic terms to a form where further simplifications or algebraic operations can be applied. In the context of solving logarithmic equations, this rule is indispensable.
2. Product Rule of Logarithms
The product rule of logarithms states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers. The mathematical representation of this rule is:
logₐ (mn) = logₐ m + logₐ n
This rule means that if you have the logarithm of a product, you can split it into the sum of the logarithms of each factor. Conversely, if you have the sum of two logarithms with the same base, you can combine them into a single logarithm of the product of their arguments.
For example, let's consider the expression log₃ (9 * 27). Using the product rule, we can rewrite this as log₃ 9 + log₃ 27. We know that log₃ 9 equals 2 (since 3² = 9) and log₃ 27 equals 3 (since 3³ = 27). Therefore, the expression simplifies to 2 + 3 = 5.
In the equation 3 log₄ x = log₄ 32 + log₄ 2, we will apply the product rule to the right side of the equation, which is log₄ 32 + log₄ 2. By using the product rule, we can combine these two terms into a single logarithm: log₄ (32 * 2), which simplifies to log₄ 64. This step is crucial for simplifying the equation and making it easier to solve.
The product rule is particularly useful in situations where dealing with individual logarithmic terms is more complex than dealing with their combined form. It allows us to reduce multiple logarithmic terms into a single term, making the equation cleaner and more manageable.
Significance of Logarithmic Properties
Understanding and being able to apply the power and product rules of logarithms are vital for solving logarithmic equations. These properties allow us to manipulate and simplify complex expressions, making them easier to work with. They are the foundational tools that turn seemingly challenging equations into manageable problems.
In our journey to solve the equation 3 log₄ x = log₄ 32 + log₄ 2, we will repeatedly use these properties. Mastery of these rules will not only help in solving this specific equation but will also equip you with the skills needed to tackle a wide range of logarithmic problems. So, let’s keep these properties in mind as we proceed with the solution.
Step-by-Step Solution to 3 log₄ x = log₄ 32 + log₄ 2
Now that we have a firm grasp of the logarithmic properties, let’s walk through the step-by-step solution to the equation:
3 log₄ x = log₄ 32 + log₄ 2
Each step will be thoroughly explained to ensure clarity and understanding. Our goal is not just to find the answer but also to understand the process, so you can confidently solve similar problems in the future.
Step 1: Apply the Product Rule of Logarithms
The first step in solving our equation is to simplify the right side by applying the product rule of logarithms. As we discussed earlier, the product rule states that logₐ (mn) = logₐ m + logₐ n. Applying this rule to our equation, we can combine the two logarithmic terms on the right side:
log₄ 32 + log₄ 2 = log₄ (32 * 2)
Performing the multiplication inside the logarithm gives us:
log₄ (32 * 2) = log₄ 64
So, our equation now looks like this:
3 log₄ x = log₄ 64
This simplification makes the equation easier to handle by reducing two logarithmic terms into one. It's a crucial step in our journey to isolate the variable x. This initial step demonstrates the power of logarithmic properties in simplifying complex equations.
Step 2: Apply the Power Rule of Logarithms
Next, we need to address the left side of the equation, which is 3 log₄ x. Here, we will use the power rule of logarithms, which states that logₐ (bⁿ) = n logₐ b. This rule allows us to move the coefficient in front of the logarithm as an exponent of the argument inside the logarithm.
Applying the power rule to 3 log₄ x, we get:
3 log₄ x = log₄ (x³)
Now our equation looks like this:
log₄ (x³) = log₄ 64
By applying the power rule, we have transformed the left side of the equation into a form that directly parallels the right side. Both sides now contain a single logarithm with the same base, which sets the stage for the next step, where we can eliminate the logarithms and equate the arguments.
Step 3: Equate the Arguments
Since we have logarithms with the same base on both sides of the equation, we can now equate the arguments. This step is based on the property that if logₐ m = logₐ n, then m = n. Applying this to our equation:
log₄ (x³) = log₄ 64
We can equate the arguments:
x³ = 64
This step simplifies the equation dramatically. We have moved from a logarithmic equation to a simple algebraic equation, which is much easier to solve. This transition is a key technique in solving logarithmic equations: using logarithmic properties to transform the equation into a more manageable form.
Step 4: Solve for x
Now we have the equation x³ = 64. To solve for x, we need to find the cube root of 64. In other words, we need to find a number that, when multiplied by itself three times, equals 64.
The cube root of 64 is 4 because:
4 * 4 * 4 = 64
So, we have:
x = ∛64 = 4
Thus, the solution to the equation is x = 4. This step involves basic algebraic manipulation, but it's crucial for isolating the variable and finding its value. We have now successfully found the solution to the simplified algebraic equation.
Step 5: Verify the Solution
It’s always a good practice to verify the solution by substituting it back into the original equation. This step ensures that our solution is correct and that we haven't made any mistakes along the way. Substituting x = 4 into the original equation:
3 log₄ x = log₄ 32 + log₄ 2
We get:
3 log₄ 4 = log₄ 32 + log₄ 2
Let’s simplify each side:
Left Side:
3 log₄ 4 = 3 * 1 = 3 (since log₄ 4 = 1)
Right Side:
We already simplified log₄ 32 + log₄ 2 to log₄ 64. Now we need to find log₄ 64. Since 4³ = 64, log₄ 64 = 3.
So, the right side is:
log₄ 64 = 3
Since both sides of the equation equal 3, our solution is verified:
3 = 3
This verification step confirms that our solution, x = 4, is correct. Verifying the solution is not just a procedural step; it's a safeguard against errors and a way to reinforce your understanding of the problem-solving process.
Conclusion
By following these steps, we have successfully solved the logarithmic equation:
3 log₄ x = log₄ 32 + log₄ 2
The solution we found is:
x = 4
This step-by-step walkthrough illustrates the importance of understanding and applying logarithmic properties, as well as the significance of verifying the solution. The combination of these skills will empower you to confidently tackle various logarithmic equations.
Key Takeaways and Further Practice
Key Takeaways
Solving logarithmic equations requires a solid understanding of logarithmic properties and careful application of algebraic techniques. Here are some key takeaways from our step-by-step solution to 3 log₄ x = log₄ 32 + log₄ 2:
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Logarithmic Properties are Essential: The product rule and power rule of logarithms are fundamental tools for simplifying equations. Being able to apply these rules correctly is crucial for transforming complex logarithmic expressions into manageable forms.
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Simplify Step-by-Step: Break down the equation into manageable steps. Simplification using logarithmic properties, followed by equating arguments, makes the problem less daunting.
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Verification is Crucial: Always verify your solution by substituting it back into the original equation. This step ensures accuracy and reinforces your understanding of the solution.
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Transform to Algebraic Form: The key to solving many logarithmic equations is to transform them into algebraic equations by using logarithmic properties. This simplifies the problem and allows for the application of familiar algebraic techniques.
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Understand the Logic: Don't just memorize steps; understand the logic behind each step. Knowing why you are applying a certain rule or technique will help you solve a wider variety of problems.
Further Practice
To reinforce your understanding and skills, try solving similar logarithmic equations. Here are a few practice problems:
- 2 log₃ x = log₃ 9 + log₃ 3
- log₂ (x²) = log₂ 16
- 4 log₅ x = log₅ 625
For each problem, follow the steps we outlined:
- Apply logarithmic properties to simplify the equation.
- Equate the arguments.
- Solve for x.
- Verify your solution.
By working through these practice problems, you'll not only solidify your understanding of logarithmic equations but also develop confidence in your problem-solving abilities. Remember, the more you practice, the more comfortable and proficient you will become.
Solving logarithmic equations is a skill that combines mathematical knowledge with problem-solving strategies. With a clear understanding of the underlying principles and consistent practice, you can master this important area of mathematics. Keep exploring, keep practicing, and you’ll find that even the most challenging equations can be solved with confidence.
