Solving Logarithmic Equations A Step By Step Guide

In the realm of mathematics, logarithmic equations often present a unique challenge, demanding a blend of algebraic manipulation and a solid understanding of logarithmic properties. Today, we embark on a journey to solve the equation $\log _6 4 x^2-\log _6 x=2$, a problem that appears deceptively simple yet requires careful execution to arrive at the correct answer. We will dissect each step, unraveling the intricacies of logarithms and showcasing the path to the solution. Our goal is not just to find the answer, but to provide a comprehensive understanding of the underlying principles, making this a valuable learning experience for anyone venturing into the world of logarithmic equations.

Deciphering the Logarithmic Puzzle

Before we dive into the specific steps for solving the equation, let's first lay the groundwork by understanding the fundamental concepts of logarithms. At its core, a logarithm is the inverse operation of exponentiation. In simpler terms, if we have an equation like $b^y = x$, then the logarithm of $x$ to the base $b$ is $y$, written as $\log_b x = y$. This means the logarithm tells us what power we need to raise the base to in order to get a certain number. Understanding this basic relationship is crucial for manipulating and solving logarithmic equations.

The Power of Logarithmic Properties

Logarithms possess several key properties that are instrumental in simplifying and solving equations. One of the most vital properties, and the one we'll be using extensively in this problem, 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: $\log_b (x/y) = \log_b x - \log_b y$. This property allows us to combine logarithmic terms that are being subtracted, which is exactly what we need to do in our equation.

Another important property is the relationship between logarithms and exponents. We know that $\log_b x = y$ is equivalent to $b^y = x$. This allows us to convert logarithmic equations into exponential equations, which are often easier to solve. We will use this property towards the end of our solution to isolate the variable $x$. Additionally, remember the fundamental definition of a logarithm: the base of the logarithm raised to the power of the result equals the argument of the logarithm. These properties are the bedrock of logarithmic manipulations and are essential tools in our problem-solving arsenal. Mastering these properties is the key to unlocking a wide range of logarithmic problems.

Step-by-Step Solution: A Journey Through Logarithms

Now, let's tackle the equation at hand: $\log _6 4 x^2-\log _6 x=2$. We will break down the solution process into manageable steps, providing clear explanations for each action we take.

Step 1: Applying the Quotient Rule

The first step in simplifying this equation is to utilize the quotient rule of logarithms. We have a difference of two logarithms with the same base, which perfectly sets the stage for applying this rule. According to the quotient rule, we can rewrite the left side of the equation as a single logarithm of a quotient: $\log _6 (\frac{4x^2}{x}) = 2$. This step condenses the equation and makes it easier to work with. Notice how the quotient rule elegantly combines the two logarithmic terms into one, a crucial simplification.

Step 2: Simplifying the Argument

Inside the logarithm, we now have the expression $\frac{4x^2}{x}$. This fraction can be simplified by canceling out a factor of $x$ from the numerator and the denominator. This yields $\frac{4x^2}{x} = 4x$, provided that $x$ is not equal to zero. Remember, the argument of a logarithm must be positive, so $x$ must be greater than zero. Our equation now looks like this: $\log _6 (4x) = 2$. This simplification makes the equation even more manageable and brings us closer to isolating the variable $x$.

Step 3: Converting to Exponential Form

To further isolate $x$, we need to convert the logarithmic equation into its equivalent exponential form. Recall the relationship between logarithms and exponents: $\log_b x = y$ is equivalent to $b^y = x$. Applying this to our equation, where the base is 6, the result is 2, and the argument is $4x$, we get $6^2 = 4x$. This transformation is a pivotal step as it removes the logarithm and allows us to work with a simple algebraic equation.

Step 4: Solving for x

Now we have the equation $6^2 = 4x$. Let's simplify this. We know that $6^2$ is equal to 36, so our equation becomes $36 = 4x$. To solve for $x$, we need to isolate it by dividing both sides of the equation by 4: $\frac{36}{4} = x$. This gives us $x = 9$. We have now arrived at a potential solution for $x$.

Step 5: Verification and the Importance of Checking Solutions

It is absolutely crucial to verify our solution in the original equation. This step is essential because logarithmic equations can sometimes produce extraneous solutions – values that satisfy the transformed equation but not the original one. Substituting $x = 9$ into the original equation, $\log _6 4 x^2-\log _6 x=2$, we get $\log _6 (4 * 9^2) - \log _6 9 = 2$. Simplifying, we have $\log _6 (4 * 81) - \log _6 9 = \log _6 324 - \log _6 9$. Applying the quotient rule again, this becomes $\log _6 (\frac{324}{9}) = \log _6 36$. Since $6^2 = 36$, we have $\log _6 36 = 2$, which matches the right side of the original equation. Therefore, $x = 9$ is indeed a valid solution. Always remember to verify your solutions in the original equation to avoid extraneous results.

The Answer and Its Significance

After careful step-by-step analysis and verification, we have arrived at the solution to the equation $\log _6 4 x^2-\log _6 x=2$. The correct answer is $x = 9$, which corresponds to option D in the given choices. This result highlights the power of logarithmic properties in simplifying complex equations. The journey we've undertaken showcases not only the mechanics of solving the problem but also the importance of understanding the underlying principles of logarithms.

Why This Matters: The Broader Implications of Logarithms

Understanding logarithmic equations is not just an academic exercise; it has far-reaching implications in various fields of science, engineering, and finance. Logarithms are used to model phenomena that span vast ranges, such as the Richter scale for earthquake magnitudes, the pH scale for acidity, and the decibel scale for sound intensity. In finance, logarithms are used in calculations involving compound interest and exponential growth. Mastering logarithmic concepts provides a valuable foundation for tackling real-world problems in these diverse domains. The ability to manipulate and solve logarithmic equations is a key skill for anyone pursuing careers in STEM fields.

Mastering Logarithms: Tips and Tricks

To truly master logarithms, consistent practice is essential. Working through a variety of problems, from basic to complex, will solidify your understanding of the properties and techniques involved. Here are some tips to help you on your journey:

• Memorize the key properties: The product rule, quotient rule, power rule, and change of base formula are your essential tools.
• Practice converting between logarithmic and exponential forms: This is a fundamental skill for solving logarithmic equations.
• Always check for extraneous solutions: This will save you from making mistakes and ensure the validity of your results.
• Break down complex problems into smaller steps: Tackle each step systematically, ensuring accuracy and clarity.
• Use online resources and textbooks: There are numerous resources available to help you learn and practice logarithms.

By following these tips and dedicating time to practice, you can develop a strong grasp of logarithms and confidently solve a wide range of logarithmic equations. Remember, the key is not just to memorize formulas but to understand the underlying concepts and their applications. Consistent practice and a deep understanding are the cornerstones of mastering logarithms.

Conclusion: The Power of Understanding

In conclusion, solving the logarithmic equation $\log _6 4 x^2-\log _6 x=2$ is a testament to the power of understanding fundamental logarithmic properties and applying them strategically. We have demonstrated a step-by-step approach, emphasizing the importance of simplification, conversion, and verification. The correct solution, $x = 9$, is a result of careful manipulation and a solid grasp of logarithmic principles. More importantly, we've highlighted the broader significance of logarithms in various fields and provided tips for mastering this crucial mathematical concept. Logarithms, while seemingly abstract, are powerful tools for understanding and modeling the world around us. By embracing the challenge and diligently practicing, you can unlock the power of logarithms and enhance your mathematical prowess.