Solving 3 Log₂(2x) = 3 A Step-by-Step Guide

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Logarithmic equations can sometimes seem daunting, but with a systematic approach, they can be solved with ease. This article will walk you through the process of solving the logarithmic equation 3 log₂(2x) = 3, providing a detailed explanation of each step to ensure clarity and understanding. We will explore the fundamental properties of logarithms and how they apply to solving such equations, ultimately leading to the correct solution.

Understanding Logarithmic Equations

At its core, a logarithmic equation involves a logarithm of an unknown variable. To effectively tackle these equations, it's crucial to grasp the basic definition of a logarithm. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Mathematically, this is expressed as logₐ(b) = c, which means aᶜ = b. Here, 'a' is the base, 'b' is the argument, and 'c' is the logarithm. Understanding this fundamental relationship is the key to unraveling logarithmic equations.

Logarithmic equations are encountered frequently in various fields, including mathematics, physics, engineering, and computer science. They are particularly useful in modeling phenomena that exhibit exponential growth or decay, such as population dynamics, radioactive decay, and compound interest. Therefore, mastering the techniques for solving logarithmic equations is not just an academic exercise but a valuable skill with real-world applications.

Before diving into the solution of the specific equation 3 log₂(2x) = 3, let's recap some essential properties of logarithms that will come in handy. These properties include the product rule, quotient rule, power rule, and the change of base formula. The power rule, which states that logₐ(bᶜ) = c logₐ(b), is particularly relevant to our problem. Also, the property that if logₐ(b) = c, then aᶜ = b, is fundamental for converting logarithmic equations into exponential form.

Step-by-Step Solution of 3 log₂(2x) = 3

Now, let's tackle the given logarithmic equation: 3 log₂(2x) = 3. Our goal is to isolate the variable 'x' and determine its value. We will proceed step-by-step, applying logarithmic properties as needed.

Step 1: Isolate the Logarithmic Term

The first step in solving the equation is to isolate the logarithmic term. This means getting the term log₂(2x) by itself on one side of the equation. To do this, we divide both sides of the equation by 3:

3 log₂(2x) / 3 = 3 / 3

This simplifies to:

log₂(2x) = 1

Now, we have the logarithmic term isolated, which makes it easier to proceed with the next step.

Step 2: Convert to Exponential Form

To eliminate the logarithm, we convert the equation from logarithmic form to exponential form. Recall that logₐ(b) = c is equivalent to aᶜ = b. In our case, the base is 2, the argument is 2x, and the logarithm is 1. Therefore, we can rewrite the equation as:

2¹ = 2x

This step is crucial because it transforms the logarithmic equation into a simpler algebraic equation that we can solve directly.

Step 3: Solve for x

Now we have a simple algebraic equation: 2 = 2x. To solve for 'x', we divide both sides of the equation by 2:

2 / 2 = 2x / 2

This simplifies to:

x = 1

So, we have found a potential solution: x = 1. However, it's essential to verify this solution to ensure it is valid.

Step 4: Verify the Solution

When dealing with logarithmic equations, it's crucial to check the solution in the original equation to make sure it doesn't lead to any undefined terms. Logarithms are only defined for positive arguments. Therefore, we need to ensure that 2x is positive when x = 1.

Substituting x = 1 into the original equation, we get:

3 log₂(2 * 1) = 3

3 log₂(2) = 3

Since log₂(2) = 1, the equation becomes:

3 * 1 = 3

3 = 3

This is a true statement, which confirms that x = 1 is indeed a valid solution. Moreover, 2x = 2 * 1 = 2, which is positive, satisfying the condition for the logarithm to be defined.

Analyzing the Given Options

Now that we have found the solution, let's analyze the given options to determine which one is correct:

A. x = -1

B. x = 1

C. x = -1 and x = 1

D. x = 0, x = -1, and x = 1

We found that x = 1 is the valid solution. Let's examine why the other options are incorrect.

If we substitute x = -1 into the original equation, we get:

3 log₂(2 * -1) = 3

3 log₂(-2) = 3

The logarithm of a negative number is undefined, so x = -1 is not a solution.

If we substitute x = 0 into the equation, we get:

3 log₂(2 * 0) = 3

3 log₂(0) = 3

The logarithm of 0 is also undefined, so x = 0 is not a solution.

Therefore, the only valid solution is x = 1, which corresponds to option B.

Common Mistakes to Avoid

When solving logarithmic equations, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and arrive at the correct solution. Here are some common mistakes to watch out for:

  1. Forgetting to check for extraneous solutions: As we saw in this example, it's crucial to check your solutions in the original equation to make sure they don't lead to undefined terms, such as the logarithm of a negative number or zero. Ignoring this step can result in incorrect answers.

  2. Incorrectly applying logarithmic properties: Logarithms have specific properties that must be applied correctly. For instance, the power rule states that logₐ(bᶜ) = c logₐ(b), but it's easy to make mistakes if you don't remember the rule accurately. Always double-check your application of logarithmic properties.

  3. Misunderstanding the domain of logarithms: Logarithms are only defined for positive arguments. This means that the expression inside the logarithm must be greater than zero. Failing to consider this restriction can lead to incorrect solutions.

  4. Making algebraic errors: Even if you understand the logarithmic concepts, algebraic errors can derail your solution. Be careful with your arithmetic and algebraic manipulations, especially when dealing with fractions, exponents, and negative signs.

  5. Skipping steps: It's tempting to skip steps to save time, but this can increase the likelihood of making errors. Write out each step clearly and logically to minimize mistakes.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving logarithmic equations.

Conclusion

In conclusion, the true solution to the logarithmic equation 3 log₂(2x) = 3 is x = 1. We arrived at this solution by isolating the logarithmic term, converting the equation to exponential form, solving for 'x', and verifying the solution in the original equation. This step-by-step approach, combined with an understanding of logarithmic properties and common pitfalls, will empower you to solve a wide range of logarithmic equations confidently.

Remember, practice is key to mastering any mathematical concept. Work through additional examples and problems to reinforce your understanding and build your problem-solving skills. With consistent effort, you'll become proficient in solving logarithmic equations and applying them to various real-world scenarios.