Solving The Logarithmic Equation Log(20x³) - 2log(x) = 4 A Step By Step Guide

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This article delves into the step-by-step solution of the logarithmic equation log(20x³) - 2log(x) = 4. We will explore the properties of logarithms, apply them strategically to simplify the equation, and ultimately arrive at the correct solution. Understanding logarithmic equations is crucial in various fields, including mathematics, physics, and engineering, making this a valuable exercise for students and professionals alike. Let's embark on this mathematical journey together, breaking down each step with clarity and precision.

Understanding Logarithmic Properties

Before we dive into solving the equation, it's essential to have a solid grasp of the fundamental logarithmic properties. These properties act as the building blocks for manipulating and simplifying logarithmic expressions. The key properties we'll utilize in this solution include:

  • The Power Rule: logₐ(xⁿ) = n logₐ(x)
  • The Quotient Rule: logₐ(x/y) = logₐ(x) - logₐ(y)
  • The Product Rule: logₐ(xy) = logₐ(x) + logₐ(y)
  • The Definition of Logarithm: If logₐ(x) = b, then aᵇ = x

These properties allow us to transform complex logarithmic expressions into simpler forms, making them easier to work with. In the context of our equation, we'll primarily use the power rule and the properties of logarithms to combine and simplify terms. Mastering these properties is the key to unlocking the solutions to a wide range of logarithmic problems. Remember, practice is essential; the more you apply these properties, the more intuitive they become.

Step-by-Step Solution

Now, let's tackle the equation log(20x³) - 2log(x) = 4. Our goal is to isolate x and find its value. We'll achieve this by systematically applying logarithmic properties and algebraic manipulations.

Step 1: Apply the Power Rule

The first step involves simplifying the term 2log(x) using the power rule. The power rule states that logₐ(xⁿ) = n logₐ(x). Applying this to our equation, we get:

log(20x³) - log(x²) = 4

This step effectively moves the coefficient 2 inside the logarithm as an exponent, which is a crucial simplification.

Step 2: Apply the Quotient Rule

Next, we can combine the two logarithmic terms using the quotient rule. The quotient rule states that logₐ(x/y) = logₐ(x) - logₐ(y). Applying this to our equation, we obtain:

log(20x³/x²) = 4

This step combines the two logarithms into a single logarithm, further simplifying the equation.

Step 3: Simplify the Argument

Now, let's simplify the argument inside the logarithm. We have 20x³/x². Dividing x³ by x² gives us x. Therefore, the equation becomes:

log(20x) = 4

This simplification makes the equation much easier to solve.

Step 4: Convert to Exponential Form

To eliminate the logarithm, we need to convert the equation from logarithmic form to exponential form. Remember the definition of a logarithm: if logₐ(x) = b, then aᵇ = x. In our case, the base of the logarithm is assumed to be 10 (since it's not explicitly written). So, we have:

10⁴ = 20x

This step transforms the logarithmic equation into a simple algebraic equation.

Step 5: Solve for x

Now, we can easily solve for x. We have 10⁴ = 20x, which means 10000 = 20x. Dividing both sides by 20, we get:

x = 10000 / 20 x = 500

Therefore, the solution to the equation is x = 500.

Verifying the Solution

It's always a good practice to verify our solution by plugging it back into the original equation. This ensures that our answer is correct and that we haven't made any errors in our calculations. Let's substitute x = 500 into the original equation:

log(20(500)³) - 2log(500) = 4

First, let's simplify the left-hand side of the equation.

log(20 * 500³) - 2log(500) = log(20 * 125000000) - 2log(500) = log(2500000000) - 2log(500)

Now, let's calculate 2log(500):

2log(500) = 2 * log(5 * 10²) = 2 * (log(5) + log(10²)) = 2 * (log(5) + 2)

We know that log(10) is 1, so log(10²) is 2. Also, log(5) is approximately 0.699. So,

2log(500) ≈ 2 * (0.699 + 2) = 2 * 2.699 ≈ 5.398

Now, let's consider log(2500000000) = log(2.5 * 10⁹) = log(2.5) + log(10⁹) = log(2.5) + 9. log(2.5) is approximately 0.3979, so

log(2500000000) ≈ 0.3979 + 9 ≈ 9.3979

Therefore, the left-hand side of the original equation is approximately:

  1. 3979 - 5.398 ≈ 4.00

This is very close to 4, which confirms that our solution x = 500 is correct. The slight discrepancy is due to rounding errors in our approximations.

Choosing the Correct Answer

Based on our step-by-step solution, we have determined that the value of x that satisfies the equation log(20x³) - 2log(x) = 4 is x = 500. Now, let's look at the given options:

A. x = 25 B. x = 50 C. x = 250 D. x = 500

Clearly, the correct answer is D. x = 500. We have not only solved the equation but also verified our solution, ensuring its accuracy.

Common Mistakes to Avoid

When solving logarithmic equations, it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:

  • Incorrect Application of Logarithmic Properties: The properties of logarithms are powerful tools, but they must be applied correctly. Make sure you understand each property thoroughly and use it appropriately.
  • Forgetting the Base of the Logarithm: When the base of the logarithm is not explicitly written, it's assumed to be 10. However, it's crucial to remember this and not treat it as if the base is 0 or 1.
  • Dividing by Zero: When simplifying equations, be careful not to divide by zero. This can lead to incorrect solutions.
  • Ignoring Extraneous Solutions: Sometimes, when solving logarithmic equations, you might obtain solutions that don't actually satisfy the original equation. These are called extraneous solutions. It's essential to verify your solutions to avoid this issue.
  • Making Arithmetic Errors: Simple arithmetic errors can derail your entire solution. Double-check your calculations to ensure accuracy.

By being aware of these common mistakes, you can significantly improve your ability to solve logarithmic equations correctly.

Practice Problems

To solidify your understanding of solving logarithmic equations, try these practice problems:

  1. Solve for x: log₂(x + 3) + log₂(x - 2) = 2
  2. Solve for x: log(x²) - log(x) = 1
  3. Solve for x: log₃(2x + 1) = 2
  4. Solve for x: log₅(x² - 4) = log₅(3x)

Working through these problems will help you hone your skills and build confidence in your ability to solve logarithmic equations. Remember, practice makes perfect!

Conclusion

In this article, we've successfully solved the logarithmic equation log(20x³) - 2log(x) = 4. We began by understanding the fundamental properties of logarithms, then systematically applied these properties to simplify the equation. We converted the equation to exponential form, solved for x, and verified our solution. The correct answer is x = 500. By following this step-by-step approach and being mindful of common mistakes, you can confidently tackle a wide range of logarithmic equations. Remember, a strong foundation in logarithmic properties is the key to success in this area of mathematics. Continue practicing, and you'll master the art of solving logarithmic equations.

Solving logarithmic equations like log(20x³) - 2log(x) = 4 requires a systematic approach, leveraging the properties of logarithms to simplify and isolate the variable. In this comprehensive guide, we'll break down each step, providing clarity and insights to ensure a thorough understanding of the solution process. This process is not only crucial for academic success but also for practical applications in various scientific and engineering fields. Let's dive into the essential steps that will help you master the art of solving logarithmic equations.

H3: Applying the Power Rule

The initial step in solving the equation log(20x³) - 2log(x) = 4 involves the strategic application of the power rule of logarithms. This rule, which states that logₐ(xⁿ) = n logₐ(x), allows us to manipulate logarithmic expressions by moving exponents from within the logarithm to the outside as coefficients, or vice versa. In our equation, we have the term 2log(x), which can be simplified using the power rule. By moving the coefficient 2 back inside the logarithm as an exponent, we transform 2log(x) into log(x²). This transformation is a crucial first step because it sets the stage for combining logarithmic terms, a key strategy in solving logarithmic equations. The application of the power rule here not only simplifies the expression but also brings us closer to a form where we can apply other logarithmic properties to further reduce the complexity of the equation. Understanding and correctly applying the power rule is fundamental to successfully navigating logarithmic problems. The ability to recognize when and how to use this rule is a hallmark of proficiency in logarithmic manipulations. By mastering this step, you lay a solid foundation for tackling more complex logarithmic challenges.

H3: Utilizing the Quotient Rule

Following the application of the power rule, the next pivotal step in solving log(20x³) - 2log(x) = 4 is the strategic use of the quotient rule of logarithms. This rule, which is expressed as logₐ(x/y) = logₐ(x) - logₐ(y), provides a powerful mechanism for combining two logarithmic expressions with the same base that are being subtracted. In our case, after applying the power rule, we have log(20x³) - log(x²) = 4. Here, we have two logarithmic terms with the same base (base 10, which is implied when no base is written) being subtracted. The quotient rule allows us to condense these two terms into a single logarithmic expression, which significantly simplifies the equation. By applying the quotient rule, we transform the left side of the equation into log(20x³/x²). This step is crucial because it reduces the number of logarithmic terms, making the equation more manageable. The quotient rule essentially reverses the process of separating a logarithm of a quotient into the difference of two logarithms, thereby streamlining the equation. Mastering the application of the quotient rule is essential for efficiently solving logarithmic equations, as it allows for the consolidation of terms and the simplification of complex expressions. This step not only simplifies the immediate equation but also prepares us for the subsequent steps in the solution process.

H3: Simplifying the Logarithmic Argument

After successfully applying the power and quotient rules, the next critical step in solving the equation log(20x³) - 2log(x) = 4 involves simplifying the argument of the resulting logarithm. Following the quotient rule, we have log(20x³/x²) = 4. The argument inside the logarithm is the expression 20x³/x², which can be significantly simplified using basic algebraic principles. When simplifying this expression, we focus on the division of the x terms. We have x³ divided by x², which results in x. Thus, the expression 20x³/x² simplifies to 20x. This simplification is a crucial step because it reduces the complexity of the logarithmic argument, making the equation easier to solve. The equation now transforms into log(20x) = 4, which is a much more manageable form compared to the original equation. The ability to simplify algebraic expressions within logarithmic arguments is a vital skill in solving logarithmic equations. This step not only makes the equation visually simpler but also brings us closer to isolating the variable x. By reducing the complexity of the argument, we prepare the equation for the next phase of the solution, which involves converting it from logarithmic to exponential form.

H3: Converting to Exponential Form

With the logarithmic argument simplified, the next crucial step in solving the equation log(20x³) - 2log(x) = 4 is to convert the equation from its logarithmic form to exponential form. This conversion is essential because it allows us to eliminate the logarithm and isolate the variable x. To convert from logarithmic to exponential form, we rely on the fundamental definition of logarithms: if logₐ(b) = c, then aᶜ = b. In our simplified equation, log(20x) = 4, the base of the logarithm is implicitly 10 (since no base is written). Applying the definition, we convert the equation to 10⁴ = 20x. This transformation is a pivotal moment in the solution process. It changes the nature of the equation from logarithmic to algebraic, making it much easier to manipulate and solve. Understanding and being able to apply the definition of logarithms is fundamental to solving logarithmic equations. This step bridges the gap between logarithmic expressions and algebraic equations, allowing us to leverage familiar algebraic techniques to find the solution. By converting to exponential form, we set the stage for the final step: isolating and solving for x.

H3: Isolating and Solving for x

Having successfully converted the equation log(20x³) - 2log(x) = 4 from logarithmic to exponential form, the final step in our solution process is to isolate and solve for the variable x. After the conversion, our equation is 10⁴ = 20x, which simplifies to 10000 = 20x. This is now a straightforward algebraic equation that we can solve using basic algebraic principles. To isolate x, we need to divide both sides of the equation by 20. Performing this division, we get x = 10000 / 20, which simplifies to x = 500. This is the solution to the equation. We have successfully isolated x and found its value. The ability to solve basic algebraic equations is essential in the final stages of solving logarithmic problems. This step demonstrates the culmination of all the previous simplifications and transformations, leading us to the final answer. By correctly isolating and solving for x, we complete the solution process and answer the original question.

H2: Conclusion: Mastering Logarithmic Equations

In conclusion, solving the logarithmic equation log(20x³) - 2log(x) = 4 demonstrates a systematic application of logarithmic properties and algebraic techniques. We began by utilizing the power rule to simplify the logarithmic terms, then applied the quotient rule to combine the logarithms. Simplifying the resulting argument, we then converted the equation to exponential form, which allowed us to isolate and solve for x. The final solution, x = 500, highlights the importance of understanding and correctly applying each step in the process. Mastering the art of solving logarithmic equations not only enhances mathematical proficiency but also provides valuable tools for various scientific and engineering applications. By practicing these techniques and understanding the underlying principles, one can confidently approach and solve a wide range of logarithmic problems.