Solving -2 Log_5(x) = Log_5(100) A Step-by-Step Guide

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In this comprehensive guide, we will tackle the equation -2 log_5(x) = log_5(100) step by step. This problem falls under the category of logarithmic equations, a fundamental topic in mathematics. Logarithmic equations appear frequently in various fields, including calculus, physics, and computer science. Therefore, mastering the techniques to solve them is crucial for anyone delving into these areas. Our discussion will not only cover the algebraic manipulations required but also provide a detailed explanation of the underlying principles and properties of logarithms. This ensures a robust understanding and equips you with the skills to approach similar problems confidently. By the end of this article, you will clearly understand how to solve logarithmic equations, particularly those involving negative coefficients and constants, and express your solutions accurately using fractions and radicals as necessary.

Before diving into the solution, let's recap the essentials of logarithmic equations. A logarithm is essentially the inverse operation of exponentiation. The expression log_b(a) = c means that b raised to the power of c equals a (b^c = a). Here, b is the base of the logarithm, a is the argument, and c is the exponent. It's crucial to remember that the base b must be positive and not equal to 1, and the argument a must be positive. These conditions ensure that the logarithmic function is well-defined. Properties of logarithms are indispensable when solving logarithmic equations. Some key properties include:

  1. Product Rule: log_b(mn) = log_b(m) + log_b(n)
  2. Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
  3. Power Rule: log_b(m^p) = p log_b(m)
  4. Change of Base Formula: log_b(a) = log_c(a) / log_c(b)

These rules allow us to manipulate logarithmic expressions, simplifying equations and making them easier to solve. The power rule, in particular, will be instrumental in tackling the given equation. Understanding these rules and how to apply them is fundamental to solving any logarithmic equation efficiently. For the equation at hand, -2 log_5(x) = log_5(100), we will primarily use the power rule to simplify the equation before isolating the variable x. Knowing the properties and constraints of logarithms allows us to approach the problem methodically and accurately.

Now, let's break down the solution to the equation -2 log_5(x) = log_5(100). We'll follow a systematic approach, explaining each step to ensure clarity and comprehension.

Step 1: Apply the Power Rule

The first step is to address the coefficient -2 in front of the logarithm. Recall the power rule: log_b(m^p) = p log_b(m). We can rewrite the left side of the equation using this rule:

-2 log_5(x) = log_5(x^-2)

Thus, the equation becomes:

log_5(x^-2) = log_5(100)

Step 2: Equate the Arguments

Since the bases of the logarithms on both sides of the equation are the same (base 5), we can equate the arguments. This step is crucial because it eliminates the logarithms, transforming the equation into a more manageable algebraic form. Therefore, we have:

x^-2 = 100

Step 3: Rewrite the Negative Exponent

To deal with the negative exponent, we rewrite x^-2 as 1/x^2. This is a fundamental property of exponents, and applying it helps simplify the equation further:

1/x^2 = 100

Step 4: Solve for x^2

Next, we want to isolate x^2. We can do this by taking the reciprocal of both sides of the equation:

x^2 = 1/100

Step 5: Solve for x

To find x, we take the square root of both sides of the equation. Remember that taking the square root yields both positive and negative solutions:

x = ±√(1/100)

Simplifying the square root gives:

x = ±(1/10)

Step 6: Check for Extraneous Solutions

In logarithmic equations, it's crucial to check for extraneous solutions. These are solutions obtained algebraically but do not satisfy the original equation's domain restrictions. Logarithms are only defined for positive arguments. In the original equation, -2 log_5(x) = log_5(100), the argument of the logarithm on the left side is x. Therefore, x must be positive.

We have two potential solutions: x = 1/10 and x = -1/10. Since x must be positive, we discard x = -1/10 as an extraneous solution.

Final Answer

The only solution that satisfies the original equation is x = 1/10. This step-by-step approach demonstrates how to solve the equation, highlighting the importance of logarithmic properties and the necessity of checking for extraneous solutions. This ensures that the final answer is mathematically sound and valid within the original equation's context. Therefore, meticulous application of these steps leads us to the correct and reliable solution.

While the step-by-step solution provided is efficient, it is worthwhile to explore alternative methods and considerations when solving logarithmic equations. This broader perspective can enhance problem-solving skills and provide a deeper understanding of logarithmic functions. Let's delve into an alternative approach and some important considerations.

Alternative Method: Direct Simplification

Instead of immediately applying the power rule, we could have approached the equation -2 log_5(x) = log_5(100) by first simplifying the right side. We know that 100 = 10^2, and since 10 = 2 * 5, we can express 100 as (2 * 5)^2. Let's rewrite the right side using this:

log_5(100) = log_5(10^2) = log_5((2 * 5)^2)

Applying the properties of logarithms, we get:

log_5((2 * 5)^2) = log_5(2^2 * 5^2) = log_5(4 * 25) = log_5(4) + log_5(25)

Since 25 = 5^2, log_5(25) = 2. Thus, we have:

log_5(100) = log_5(4) + 2

Now, our original equation becomes:

-2 log_5(x) = log_5(4) + 2

Divide both sides by -2:

log_5(x) = -1/2 log_5(4) - 1

Applying the power rule in reverse:

log_5(x) = log_5(4^(-1/2)) - 1

We know that 4^(-1/2) = 1/√4 = 1/2, so:

log_5(x) = log_5(1/2) - 1

Rewrite 1 as log_5(5):

log_5(x) = log_5(1/2) - log_5(5)

Using the quotient rule:

log_5(x) = log_5((1/2) / 5) = log_5(1/10)

Equating the arguments, we get:

x = 1/10

This alternative method, while more complex, demonstrates the flexibility in using logarithmic properties to solve equations. It reinforces the idea that there may be multiple paths to the solution, each requiring careful application of logarithmic rules.

Important Considerations

  1. Domain Restrictions: Always remember that the arguments of logarithms must be positive. This is a crucial consideration when solving logarithmic equations. Failure to check for extraneous solutions can lead to incorrect answers. In our example, we correctly discarded x = -1/10 because it would result in taking the logarithm of a negative number, which is undefined.

  2. Base Consistency: When equating logarithmic expressions, ensure that the bases are the same. If the bases are different, you may need to use the change of base formula to make them consistent.

  3. Simplification Strategies: There is often more than one way to simplify a logarithmic equation. Choosing the most efficient method can save time and reduce the chances of making errors. In the alternative method discussed, simplifying the constant term first provided a different pathway to the solution.

  4. Use of Technology: Calculators and software tools can be helpful in solving complex logarithmic equations, especially when dealing with approximations or verifying solutions. However, it's essential to understand the underlying mathematical principles to interpret the results correctly.

By considering these alternative methods and important factors, one can approach logarithmic equations with greater confidence and versatility. This holistic approach not only enhances problem-solving skills but also deepens the understanding of logarithmic functions and their properties. Embracing different perspectives and leveraging various strategies ultimately contributes to mathematical proficiency and adaptability in diverse problem-solving contexts.

Solving logarithmic equations can be tricky, and certain common mistakes can lead to incorrect solutions. Recognizing these pitfalls is crucial for achieving accuracy and building a solid understanding of logarithmic functions. Let's explore some frequent errors and how to avoid them.

1. Neglecting Domain Restrictions

One of the most common mistakes is forgetting that the argument of a logarithm must be positive. This restriction is fundamental to the definition of logarithms. For example, in the equation -2 log_5(x) = log_5(100), we found potential solutions x = 1/10 and x = -1/10. However, only x = 1/10 is valid because the logarithm of a negative number is undefined. Failing to check this condition can lead to extraneous solutions, which are algebraically correct but not valid in the original equation.

How to Avoid: Always check your solutions by plugging them back into the original equation. Ensure that the arguments of all logarithms are positive. If a solution results in a non-positive argument, it must be discarded.

2. Incorrect Application of Logarithmic Properties

Logarithmic properties, such as the product rule, quotient rule, and power rule, are powerful tools for simplifying equations. However, they must be applied correctly. A common mistake is misapplying these rules or applying them in the wrong context. For instance, the product rule states that log_b(mn) = log_b(m) + log_b(n), but it's incorrect to assume that log_b(m + n) equals log_b(m) + log_b(n). Similarly, the power rule, log_b(m^p) = p log_b(m), should not be confused with (log_b(m))^p.

How to Avoid: Review and understand each logarithmic property thoroughly. Practice applying them in various contexts to build familiarity. When manipulating logarithmic expressions, double-check that you are using the correct rule and applying it appropriately.

3. Incorrectly Equating Arguments

When you have an equation of the form log_b(m) = log_b(n), you can equate the arguments, resulting in m = n. However, this is only valid if the logarithms have the same base. Attempting to equate arguments when the bases differ is a critical error. For example, if you have log_2(x) = log_3(5), you cannot simply say x = 5.

How to Avoid: Before equating arguments, ensure that the bases of the logarithms are identical. If they are not, use the change of base formula to convert them to a common base.

4. Algebraic Errors

Solving logarithmic equations often involves algebraic manipulations, such as simplifying expressions, solving for variables, and taking square roots. Errors in these steps can lead to incorrect solutions. For instance, when solving x^2 = 1/100, it's essential to remember that taking the square root yields both positive and negative solutions (x = ±1/10). Neglecting the negative solution can result in an incomplete answer.

How to Avoid: Pay close attention to detail during each algebraic step. Double-check your calculations and ensure you have considered all possible solutions. Practice algebraic manipulations to improve your accuracy and speed.

5. Ignoring the Definition of Logarithms

The very definition of a logarithm is a potential pitfall if not fully understood. The expression log_b(a) = c means b^c = a. Misunderstanding this fundamental relationship can lead to errors in solving equations. For example, when dealing with equations involving exponents and logarithms, ensure you correctly convert between these forms.

How to Avoid: Reinforce your understanding of the definition of logarithms and their inverse relationship with exponential functions. Practice converting between logarithmic and exponential forms to solidify your knowledge.

By being aware of these common mistakes and actively working to avoid them, you can significantly improve your accuracy and confidence in solving logarithmic equations. Consistent practice, attention to detail, and a solid understanding of logarithmic properties are key to mastering this essential mathematical concept.

In this comprehensive guide, we have thoroughly explored the process of solving the logarithmic equation -2 log_5(x) = log_5(100). We began by establishing a clear understanding of logarithmic equations and their fundamental properties, which laid the groundwork for our step-by-step solution. The solution involved applying the power rule, equating arguments, rewriting exponents, and solving for x. Crucially, we emphasized the importance of checking for extraneous solutions, a step often overlooked but vital for ensuring the correctness of the final answer. Through this methodical approach, we determined that the exact solution to the equation is x = 1/10.

We further enhanced our understanding by examining an alternative method for solving the equation. This alternative approach showcased the versatility of logarithmic properties and highlighted the fact that multiple paths can lead to the correct solution. This reinforces the idea that mathematical problem-solving is not about finding a single solution but about developing a flexible and adaptable mindset.

Additionally, we addressed common mistakes that students and practitioners often encounter when solving logarithmic equations. These include neglecting domain restrictions, incorrectly applying logarithmic properties, and making algebraic errors. By recognizing these potential pitfalls and implementing strategies to avoid them, we can significantly improve our accuracy and efficiency in solving logarithmic equations.

In summary, mastering the solution of logarithmic equations requires a combination of understanding logarithmic properties, careful algebraic manipulation, and a keen awareness of potential errors. The step-by-step approach, alternative methods, and common mistakes discussed in this guide provide a robust framework for tackling a wide range of logarithmic problems. By consistently practicing these techniques and reinforcing your understanding of logarithmic principles, you can confidently solve complex equations and apply these skills in various mathematical and scientific contexts. The journey through this equation has not only provided a solution but also a deeper appreciation for the elegance and power of logarithmic functions in mathematics.