Solving Logarithmic Equations A Step-by-Step Guide To Solve Log₂(x) - Log₂(3) = 2

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In the realm of mathematics, logarithmic equations play a crucial role in modeling various phenomena, ranging from exponential growth and decay to the measurement of sound intensity and earthquake magnitudes. Understanding how to solve these equations is essential for students, engineers, scientists, and anyone dealing with quantitative analysis. This article delves into the step-by-step process of solving a specific logarithmic equation: log₂(x) - log₂(3) = 2. We will explore the fundamental properties of logarithms, apply these properties to simplify the equation, and ultimately determine the value of x that satisfies the given condition. This comprehensive guide aims to provide a clear and concise explanation, making it accessible to individuals with varying levels of mathematical background.

Before we dive into solving the equation, let's briefly revisit the concept of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, if we have an equation of the form by = x, then the logarithm of x to the base b is y, written as logb(x) = y. Here,

  • b is the base of the logarithm (a positive number not equal to 1).
  • x is the argument of the logarithm (a positive number).
  • y is the exponent to which the base must be raised to obtain x.

Key Properties of Logarithms: Understanding the properties of logarithms is crucial for simplifying and solving logarithmic equations.

  • Product Rule: logb(mn) = logb(m) + logb(n) - The logarithm of the product of two numbers is equal to the sum of their logarithms.
  • Quotient Rule: logb(m/ n) = logb(m) - logb(n) - The logarithm of the quotient of two numbers is equal to the difference of their logarithms.
  • Power Rule: logb(mp) = p logb(m) - The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.
  • Change of Base Rule: loga(x) = logb(x) / logb(a) - This rule allows you to change the base of a logarithm, which is particularly useful when using calculators that only have common logarithms (base 10) or natural logarithms (base e).

In our equation, log₂(x) - log₂(3) = 2, we are dealing with logarithms to the base 2. These properties will be essential in simplifying the equation and isolating x.

Now, let's tackle the equation step by step. The equation we need to solve is:

log₂(x) - log₂(3) = 2

Step 1: Apply the Quotient Rule of Logarithms

The first step in solving this equation is to recognize that we have a difference of two logarithms with the same base. According to the quotient rule of logarithms, the difference of two logarithms with the same base is equal to the logarithm of the quotient of their arguments. Therefore, we can rewrite the equation as:

log₂(x/3) = 2

This step simplifies the equation by combining the two logarithmic terms into a single term. This single logarithm is much easier to manipulate and solve for x. Applying logarithmic rules correctly is critical in these types of problems. The quotient rule allows us to condense the left-hand side of the equation, bringing us closer to isolating x.

Step 2: Convert the Logarithmic Equation to Exponential Form

To further simplify the equation and isolate x, we need to convert the logarithmic equation into its equivalent exponential form. Recall that a logarithm is the inverse of an exponential function. The general relationship between logarithms and exponents is:

logb(x) = y is equivalent to by = x

In our case, we have log₂(x/3) = 2. Applying the definition, we can rewrite this as:

2² = x/3

This conversion eliminates the logarithm, transforming the equation into a simple algebraic form. This step is crucial because it allows us to work with familiar algebraic operations to solve for x. Understanding the relationship between logarithmic and exponential forms is fundamental to solving logarithmic equations.

Step 3: Solve for x

Now that we have converted the equation to exponential form, we can solve for x. The equation is:

2² = x/3

First, calculate 2²:

4 = x/3

Next, to isolate x, we multiply both sides of the equation by 3:

4 * 3 = (x/3) * 3

12 = x

Thus, we have found the value of x that satisfies the original equation. Solving for x involves basic algebraic manipulation, which is made straightforward once the logarithmic equation has been converted to exponential form. The final step confirms our solution by isolating the variable.

Step 4: Verify the Solution

It is always a good practice to verify the solution by plugging the value of x back into the original equation to ensure it holds true. The original equation is:

log₂(x) - log₂(3) = 2

Substitute x = 12:

log₂(12) - log₂(3) = 2

Apply the quotient rule of logarithms:

log₂(12/3) = 2

log₂(4) = 2

Since 2² = 4, the equation holds true.

This verification step is essential to confirm that our solution is correct and that no errors were made during the solving process. By plugging the value back into the original equation, we ensure that the logarithmic relationship is maintained.

When solving logarithmic equations, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them:

  1. Incorrectly Applying Logarithmic Properties: The logarithmic properties, such as the product rule, quotient rule, and power rule, are crucial for simplifying logarithmic expressions and equations. However, these properties must be applied correctly. For instance, it is a common mistake to assume that logb(m - n) is equal to logb(m) - logb(n), which is incorrect. The correct rule applies to the logarithm of a quotient, not a difference. Always double-check the properties before applying them.

  2. Forgetting to Verify Solutions: Logarithmic equations can sometimes yield extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. This often happens when dealing with equations that involve squaring or other operations that can introduce additional roots. To avoid this, always verify your solutions by substituting them back into the original equation. If a solution does not satisfy the original equation, it is an extraneous solution and should be discarded.

  3. Ignoring the Domain of Logarithmic Functions: The domain of a logarithmic function is restricted to positive real numbers. In other words, the argument of a logarithm must be greater than zero. When solving logarithmic equations, it is essential to consider the domain restrictions. If a solution results in taking the logarithm of a non-positive number, it is not a valid solution. Always check that your solutions are within the domain of the logarithmic functions involved in the equation.

  4. Misunderstanding the Change of Base Formula: The change of base formula is a valuable tool for evaluating logarithms with bases that are not directly available on a calculator. However, it is important to use the formula correctly. A common mistake is to invert the numerator and denominator or to apply the formula incorrectly. Ensure you understand the formula and practice using it to avoid errors.

  5. Algebraic Errors: As with any mathematical problem, algebraic errors can lead to incorrect solutions. Mistakes in arithmetic, such as adding, subtracting, multiplying, or dividing terms, can propagate through the solution process. Be careful with your algebra and double-check each step to minimize the risk of errors.

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

In conclusion, solving the logarithmic equation log₂(x) - log₂(3) = 2 involves understanding and applying the properties of logarithms, converting the equation to exponential form, and solving for the unknown variable. We successfully found that x = 12 is the solution to the equation. By following a systematic approach and verifying the solution, we can ensure the accuracy of our results. Logarithmic equations are a fundamental part of mathematics, with applications in various fields, making it essential to master the techniques for solving them.

This article has provided a comprehensive guide to solving logarithmic equations, emphasizing the importance of logarithmic properties, algebraic manipulation, and verification. By understanding these concepts, you can confidently tackle more complex logarithmic problems and apply them in real-world scenarios.