Solving 2ln(5x) = -2 Step-by-Step Guide

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In the realm of mathematics, solving equations is a fundamental skill. It allows us to unravel the unknown and discover the values that satisfy specific conditions. Among various types of equations, logarithmic equations often pose a unique challenge. This article delves into the process of solving a logarithmic equation, providing a step-by-step guide and a thorough explanation of the underlying concepts. We will specifically focus on the equation 2ln(5x) = -2, demonstrating how to isolate the variable x and arrive at the solution.

Understanding Logarithmic Equations

Before we embark on solving the equation, it's essential to grasp the essence of logarithmic equations. A logarithmic equation is an equation where the logarithm of an expression containing a variable is involved. The logarithm, denoted as "log," is the inverse operation of exponentiation. In simpler terms, if we have an equation like b^y = x, the logarithm of x to the base b is y, written as log_b(x) = y. The most common logarithm is the natural logarithm, denoted as "ln," which has a base of e (Euler's number, approximately 2.71828).

To effectively solve logarithmic equations, it is crucial to understand the properties of logarithms. These properties allow us to manipulate and simplify equations, ultimately leading us to the solution. Some key properties include:

  • Product Rule: log_b(mn) = log_b(m) + log_b(n)
  • Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
  • Power Rule: log_b(m^p) = p * log_b(m)
  • Inverse Property: b^(log_b(x)) = x and log_b(b^x) = x

With a firm understanding of logarithmic equations and their properties, we can now proceed to solve the given equation.

Step-by-Step Solution for 2ln(5x) = -2

Let's break down the process of solving the equation 2ln(5x) = -2 into manageable steps.

Step 1: Isolate the Logarithmic Term

The initial step involves isolating the logarithmic term, ln(5x), on one side of the equation. To achieve this, we divide both sides of the equation by 2:

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

Now we have the equation in a more simplified form, where the logarithmic term is isolated.

Step 2: Convert to Exponential Form

To eliminate the logarithm, we need to convert the equation from logarithmic form to exponential form. Recall that the natural logarithm has a base of e. Therefore, the equation ln(5x) = -1 can be rewritten in exponential form as:

e^(-1) = 5x

This step is crucial as it allows us to remove the logarithm and work with a more familiar exponential equation.

Step 3: Isolate the Variable x

Our next goal is to isolate the variable x. To do this, we divide both sides of the equation by 5:

e^(-1) / 5 = 5x / 5
x = e^(-1) / 5

Now we have x expressed in terms of e^(-1).

Step 4: Calculate the Value of x

To obtain a numerical value for x, we need to calculate e^(-1) and divide the result by 5. Recall that e^(-1) is the same as 1/e. The approximate value of e is 2.71828.

x = (1 / e) / 5
x ≈ (1 / 2.71828) / 5
x ≈ 0.367879 / 5
x ≈ 0.073575

Step 5: Round to the Nearest Tenth

The problem statement requires us to round the answer to the nearest tenth. Therefore, we round 0.073575 to 0.1.

x ≈ 0.1

Therefore, the solution to the equation 2ln(5x) = -2, rounded to the nearest tenth, is approximately x = 0.1.

Alternative Method: Using Logarithm Properties

While the previous method provides a direct approach to solving the equation, we can also employ logarithm properties to simplify the equation before converting it to exponential form. This alternative method showcases the versatility of logarithm properties in problem-solving.

Step 1: Apply the Power Rule

We begin by applying the power rule of logarithms, which states that log_b(m^p) = p * log_b(m). In our equation, 2ln(5x) = -2, we can rewrite the left side using the power rule:

ln((5x)^2) = -2
ln(25x^2) = -2

By applying the power rule, we have transformed the equation into a different form while maintaining its equivalence.

Step 2: Convert to Exponential Form

Now, we convert the equation from logarithmic form to exponential form, similar to the previous method. Since we have the natural logarithm (base e), we can rewrite the equation as:

e^(-2) = 25x^2

This step is essential for eliminating the logarithm and working with an exponential equation.

Step 3: Isolate the Variable x^2

To isolate x^2, we divide both sides of the equation by 25:

e^(-2) / 25 = 25x^2 / 25
x^2 = e^(-2) / 25

We now have x^2 isolated on one side of the equation.

Step 4: Take the Square Root

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

x = ±√(e^(-2) / 25)

Since the domain of the natural logarithm function requires the argument to be positive (5x > 0, implying x > 0), we only consider the positive root. This is a critical step to ensure we adhere to the constraints of logarithmic functions.

x = √(e^(-2) / 25)

Step 5: Simplify and Calculate the Value of x

We can simplify the expression further:

x = √(1 / (25e^2))
x = 1 / (5e)

Now, we calculate the value of x:

x ≈ 1 / (5 * 2.71828)
x ≈ 1 / 13.5914
x ≈ 0.073575

Step 6: Round to the Nearest Tenth

Rounding to the nearest tenth, we get:

x ≈ 0.1

As expected, we arrive at the same solution, x ≈ 0.1, using this alternative method. This demonstrates the flexibility and power of logarithmic properties in solving equations.

Key Considerations and Potential Pitfalls

When solving logarithmic equations, it's crucial to be mindful of certain considerations and potential pitfalls. Ignoring these aspects can lead to incorrect solutions or missed solutions.

1. Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive arguments. This means that the expression inside the logarithm must be greater than zero. In our original equation, 2ln(5x) = -2, the argument is 5x. Therefore, we must have 5x > 0, which implies x > 0. This condition is crucial for determining the validity of any solutions we obtain.

2. Extraneous Solutions

When solving logarithmic equations, it's possible to obtain solutions that do not satisfy the original equation. These are called extraneous solutions. Extraneous solutions often arise when we manipulate the equation, such as converting it to exponential form or squaring both sides. Therefore, it's essential to check all solutions in the original equation to ensure they are valid.

3. Base of the Logarithm

Pay close attention to the base of the logarithm. If no base is explicitly written, it is usually assumed to be base 10 (common logarithm) or base e (natural logarithm). Using the correct base is critical for accurate calculations and conversions.

4. Properties of Logarithms

A thorough understanding of logarithm properties is indispensable for solving logarithmic equations. Incorrectly applying or misunderstanding these properties can lead to errors. Regularly review and practice using the properties to solidify your understanding.

Conclusion

Solving logarithmic equations requires a systematic approach and a solid understanding of logarithmic principles. In this article, we meticulously solved the equation 2ln(5x) = -2 using two different methods, both yielding the solution x ≈ 0.1. We also highlighted the importance of considering the domain of logarithmic functions and checking for extraneous solutions. By mastering these techniques and concepts, you can confidently tackle a wide range of logarithmic equations.

Remember, practice is key to proficiency in mathematics. Work through various examples and gradually increase the complexity of the equations you solve. With dedication and perseverance, you can unlock the power of logarithms and excel in your mathematical endeavors. The ability to solve logarithmic equations is not only a valuable mathematical skill but also a powerful tool for tackling problems in various scientific and engineering disciplines. So, embrace the challenge and embark on your journey to logarithmic mastery!