Solving Logarithmic Equations A Step-by-Step Guide To Log₂(9x) - Log₂3 = 3

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In this comprehensive guide, we will delve into the step-by-step solution of the logarithmic equation log₂(9x) - log₂3 = 3. Logarithmic equations might seem intimidating at first, but with a clear understanding of logarithmic properties and algebraic manipulation, they become quite manageable. This article aims to provide you with a detailed, easy-to-follow approach to solve this specific equation, while also enhancing your general problem-solving skills in mathematics. We will break down each step, explaining the reasoning and the mathematical principles involved, ensuring that you grasp not just the solution, but also the underlying concepts. This detailed exploration will be beneficial for students, educators, and anyone with an interest in mathematics. Understanding how to solve such equations is crucial as logarithmic functions are widely used in various fields like physics, engineering, computer science, and finance. By the end of this article, you will not only be able to solve this particular problem but will also have gained the confidence to tackle similar logarithmic challenges.

Understanding Logarithmic Properties

Before we dive into solving the equation log₂(9x) - log₂3 = 3, it’s crucial to understand the fundamental properties of logarithms. Logarithms are essentially the inverse operation of exponentiation. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. For instance, log₂8 = 3 because 2 raised to the power of 3 equals 8 (2³ = 8). Grasping this basic definition is essential for understanding and manipulating logarithmic expressions. One of the most important properties we will use in solving this equation is the quotient rule of logarithms. This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Mathematically, this is expressed as logₐ(b/c) = logₐ(b) - logₐ(c), where 'a' is the base, and 'b' and 'c' are the numerator and denominator, respectively. This property is a cornerstone in simplifying logarithmic expressions and is particularly relevant in our given equation where we have a difference of two logarithms. Another crucial concept is the relationship between logarithms and exponential forms. The logarithmic equation logₐ(b) = c can be rewritten in exponential form as aᶜ = b. This transformation is often key to solving logarithmic equations, as it allows us to move from the logarithmic domain to the more familiar exponential domain. We will utilize this property to eliminate the logarithm from our equation and solve for the unknown variable, 'x'.

Step-by-Step Solution

Let’s tackle the equation log₂(9x) - log₂3 = 3 step by step. Our primary goal is to isolate 'x', and we'll achieve this by strategically applying logarithmic properties and algebraic manipulations. First, we need to simplify the left-hand side of the equation. Notice that we have a difference of two logarithms with the same base (base 2). This is where the quotient rule of logarithms comes into play. Recall that logₐ(b) - logₐ(c) = logₐ(b/c). Applying this rule to our equation, we can combine the two logarithms into a single logarithm: log₂(9x/3) = 3. Now, we simplify the fraction inside the logarithm. 9x divided by 3 simplifies to 3x, so our equation becomes: log₂(3x) = 3. We've successfully reduced the equation to a simpler form with a single logarithm. The next crucial step is to eliminate the logarithm. To do this, we'll convert the logarithmic equation into its equivalent exponential form. Remember that logₐ(b) = c is equivalent to aᶜ = b. Applying this to our equation log₂(3x) = 3, we get 2³ = 3x. This transformation is a key move in solving logarithmic equations as it allows us to work with a more familiar algebraic form. Now, we can easily solve for 'x'. 2³ equals 8, so our equation is now 8 = 3x. To isolate 'x', we divide both sides of the equation by 3: x = 8/3. Therefore, the solution to the equation log₂(9x) - log₂3 = 3 is x = 8/3. This step-by-step process highlights the importance of understanding and applying logarithmic properties to simplify and solve logarithmic equations effectively.

Verification of the Solution

After finding a solution to any equation, especially logarithmic equations, it's crucial to verify the solution. This step ensures that our solution is correct and that we haven't introduced any extraneous solutions. Extraneous solutions can arise in logarithmic equations because the domain of a logarithmic function is restricted to positive numbers. Plugging an extraneous solution back into the original equation might lead to taking the logarithm of a negative number or zero, which is undefined. To verify our solution, x = 8/3, we substitute this value back into the original equation: log₂(9x) - log₂3 = 3. Replace 'x' with 8/3: log₂(9*(8/3)) - log₂3 = 3. Simplify inside the first logarithm: 9 * (8/3) equals 24, so we have log₂(24) - log₂3 = 3. Now, we can use the quotient rule of logarithms in reverse. Recall that logₐ(b) - logₐ(c) = logₐ(b/c). Applying this rule, we combine the two logarithms: log₂(24/3) = 3. Simplify the fraction: 24/3 equals 8, so we have log₂8 = 3. This statement is true because 2 raised to the power of 3 equals 8 (2³ = 8). Since our solution, x = 8/3, satisfies the original equation, we can confidently conclude that it is a valid solution. This verification step underscores the importance of checking solutions in logarithmic equations to ensure accuracy and avoid extraneous results.

Common Mistakes to Avoid

When solving logarithmic equations, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls is crucial for achieving accuracy. One frequent error is misapplying the logarithmic properties. For instance, students sometimes mistakenly apply the quotient rule to expressions that do not fit the required form, such as trying to simplify logₐ(b) / logₐ(c) as logₐ(b/c), which is incorrect. The quotient rule applies to the difference of logarithms, not the quotient. Another common mistake is neglecting to verify the solution. As mentioned earlier, logarithmic functions are only defined for positive arguments. Therefore, any solution that results in taking the logarithm of a negative number or zero is extraneous and must be discarded. Always substitute your solution back into the original equation to ensure it is valid. A third mistake arises from incorrectly converting between logarithmic and exponential forms. It’s essential to remember the fundamental relationship: logₐ(b) = c is equivalent to aᶜ = b. Confusing the base, exponent, or result in this conversion can lead to incorrect solutions. Finally, students often make algebraic errors in the simplification process. These can include mistakes in arithmetic, such as incorrect multiplication or division, or errors in applying the order of operations. Careful attention to detail and thorough checking of each step are essential to avoid these errors. By being mindful of these common mistakes and taking steps to prevent them, you can significantly improve your accuracy in solving logarithmic equations. Regular practice and a solid understanding of the fundamental principles are key to mastering these types of problems.

Conclusion

In summary, solving the logarithmic equation log₂(9x) - log₂3 = 3 involves a series of logical steps grounded in the properties of logarithms. We began by understanding the core principles of logarithms, particularly the quotient rule, which allows us to combine logarithmic expressions. This led us to simplify the equation to log₂(3x) = 3. The next critical step was converting the logarithmic equation into its exponential form, 2³ = 3x, which enabled us to isolate 'x'. Through simple algebraic manipulation, we found the solution to be x = 8/3. However, the solution process doesn't end here. We emphasized the importance of verifying the solution by substituting it back into the original equation. This step ensures that our solution is valid and not an extraneous result, a common issue in logarithmic equations. We also discussed common mistakes to avoid, such as misapplying logarithmic properties, neglecting verification, incorrectly converting between logarithmic and exponential forms, and making algebraic errors. By understanding these pitfalls, you can improve your accuracy and problem-solving skills in mathematics. The ability to solve logarithmic equations is not just an academic exercise; it has practical applications in various fields. Therefore, mastering these techniques is a valuable skill. We hope this comprehensive guide has provided you with a clear understanding of how to solve this specific equation and has equipped you with the knowledge to tackle similar logarithmic challenges with confidence.