Solving Logarithmic Equations 2 Log₅ X = Log₅ 4

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In the realm of mathematics, logarithmic equations present a unique challenge and a fascinating area of exploration. This article delves into the process of solving the logarithmic equation 2 log₅ x = log₅ 4, providing a step-by-step guide to arrive at the correct solution(s). We will also analyze the potential pitfalls and common mistakes that students often encounter while tackling such problems. This exploration will not only help you understand the specific equation but also equip you with the skills to solve a wide array of logarithmic problems.

Understanding Logarithmic Equations

Before we dive into the specifics of the equation 2 log₅ x = log₅ 4, it's crucial to have a solid grasp of the fundamental concepts of logarithms. A logarithm is essentially 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. Understanding this relationship is paramount to solving logarithmic equations effectively. The base of a logarithm (in this case, 5) is the number that is raised to a power, and the argument (x in log₅ x) is the value for which we are trying to find the exponent. Logarithmic equations often require us to manipulate these relationships to isolate the variable and find its value. Remember, the argument of a logarithm must always be positive, as logarithms are not defined for non-positive numbers. This is a critical point to keep in mind when checking for extraneous solutions later on.

The properties of logarithms are the key tools we'll use to solve equations. One of the most important properties, and the one we'll use extensively in this case, is the power rule. The power rule states that log_b (x^p) = p log_b x. This rule allows us to move exponents inside a logarithm as coefficients outside the logarithm and vice versa. Another fundamental property is the one-to-one property, which states that if log_b x = log_b y, then x = y. This property is extremely useful for simplifying equations where we have logarithms with the same base on both sides. Additionally, recall the definition of a logarithm: log_b x = y is equivalent to b^y = x. This definition can be used to convert logarithmic equations into exponential form, which can sometimes make them easier to solve. Finally, understanding the domain of logarithmic functions is crucial. The argument of a logarithm must always be positive, so when solving logarithmic equations, we must check our solutions to ensure they do not result in taking the logarithm of a non-positive number.

Step-by-Step Solution of 2 log₅ x = log₅ 4

Let's break down the process of solving the equation 2 log₅ x = log₅ 4 into manageable steps. This methodical approach will help ensure accuracy and clarity in your solution.

  1. Apply the Power Rule of Logarithms: The first step in solving this equation is to utilize the power rule of logarithms. This rule allows us to rewrite the left side of the equation, 2 log₅ x, as log₅ (x²). By applying this rule, we effectively eliminate the coefficient 2 in front of the logarithm. This simplifies the equation and prepares it for the next step. The equation now becomes log₅ (x²) = log₅ 4. This transformation is crucial because it allows us to directly compare the arguments of the logarithms on both sides of the equation.
  2. Use the One-to-One Property: Now that we have a single logarithm on each side of the equation with the same base (5), we can apply the one-to-one property of logarithms. This property states that if log_b x = log_b y, then x = y. In our case, this means that if log₅ (x²) = log₅ 4, then x² = 4. This step significantly simplifies the equation, transforming it from a logarithmic equation into a simple algebraic equation. We have effectively eliminated the logarithms and are now dealing with a quadratic equation, which is much easier to solve.
  3. Solve the Quadratic Equation: The equation x² = 4 is a classic quadratic equation. To solve it, we take the square root of both sides. Remember that when taking the square root, we must consider both the positive and negative roots. This gives us two possible solutions: x = 2 and x = -2. These are the potential solutions to our original logarithmic equation, but we must verify them to ensure they are valid.
  4. Check for Extraneous Solutions: This is a critical step in solving logarithmic equations. Since the logarithm of a non-positive number is undefined, we must check if our potential solutions make the argument of the logarithm positive. In the original equation, 2 log₅ x = log₅ 4, the argument of the logarithm on the left side is x. If we substitute x = -2, we get 2 log₅ (-2), which is undefined because we cannot take the logarithm of a negative number. Therefore, x = -2 is an extraneous solution and must be discarded. On the other hand, if we substitute x = 2, we get 2 log₅ (2), which is a valid expression. Thus, x = 2 is a valid solution. After performing this check, we can confidently state that the only solution to the equation is x = 2.

Identifying the Correct Solution

After meticulously following the steps outlined above, we arrive at the conclusion that the only valid solution for the equation 2 log₅ x = log₅ 4 is x = 2. Let's revisit the initial options provided:

  • x = -2: As we discussed, this is an extraneous solution because it results in taking the logarithm of a negative number.
  • x = 4: Substituting x = 4 into the original equation gives us 2 log₅ 4 = log₅ 4, which simplifies to log₅ 16 = log₅ 4. This is not true, so x = 4 is not a solution.
  • x = 16: If we substitute x = 16 into the original equation, we get 2 log₅ 16 = log₅ 4, which simplifies to log₅ 256 = log₅ 4. This is also not true, so x = 16 is not a solution.
  • x = -10: This is another extraneous solution because it would result in taking the logarithm of a negative number.
  • x = 2: This is the correct solution, as we have demonstrated through our step-by-step solution and verification process.

Common Mistakes and Pitfalls

Solving logarithmic equations can be tricky, and there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your problem-solving accuracy. One of the most frequent errors is forgetting to check for extraneous solutions. As we've seen, not all solutions obtained algebraically are valid solutions to the original logarithmic equation. It is crucial to always substitute your potential solutions back into the original equation and ensure that they do not result in taking the logarithm of a non-positive number. Another common mistake is misapplying the properties of logarithms. For example, students might incorrectly try to combine logarithms with different bases or apply the power rule in the wrong way. It is essential to thoroughly understand and correctly apply the logarithmic properties. A further pitfall is incorrectly solving the resulting algebraic equation after applying the logarithmic properties. For instance, in our example, after applying the one-to-one property, we obtained the quadratic equation x² = 4. It's crucial to remember to consider both the positive and negative roots when solving such equations. By avoiding these common mistakes and diligently following the steps outlined in this article, you can confidently solve a wide variety of logarithmic equations.

Conclusion

In conclusion, solving the logarithmic equation 2 log₅ x = log₅ 4 requires a clear understanding of logarithmic properties, a methodical approach, and careful attention to detail. By applying the power rule, the one-to-one property, and checking for extraneous solutions, we have successfully determined that the only valid solution is x = 2. Remember, logarithmic equations are a fundamental part of mathematics, and mastering them requires practice and a solid grasp of the underlying principles. By understanding the concepts discussed in this article and avoiding common pitfalls, you can confidently tackle logarithmic problems and excel in your mathematical endeavors.