Solving Logarithmic Equations With The One-to-One Property Log Base 2 (2x) = Log Base 2 (16)

Jul 10, 2025 by ADMIN 93 views

Solve Logarithmic Equations Using the One-to-One Property: A Step-by-Step Guide

In the realm of mathematics, logarithmic equations often present a unique challenge. However, with a solid understanding of logarithmic properties, these equations can be solved systematically. One powerful technique is leveraging the one-to-one property of logarithms, which provides a direct approach to isolating the variable. In this comprehensive guide, we will delve into the intricacies of solving the equation $\log _2 2 x=\log _2 16$ using this property, providing a step-by-step explanation and highlighting the underlying principles. We will also explore the significance of the one-to-one property and its broader applications in solving logarithmic equations.

Understanding the One-to-One Property of Logarithms

The cornerstone of our solution lies in the one-to-one property of logarithms. This property states that if we have an equation where the logarithms of two expressions are equal and share the same base, then the expressions themselves must be equal. Mathematically, this can be expressed as follows:

If $\log_b m = \log_b n$ , then $m = n$ , provided that $b > 0$ , $b \neq 1$ , $m > 0$ , and $n > 0$ .

This property is a direct consequence of the fact that logarithmic functions are one-to-one, meaning that each input corresponds to a unique output, and vice versa. This uniqueness allows us to eliminate the logarithmic functions and focus on the arguments within them.

Before applying the one-to-one property, it's crucial to ensure that the logarithms on both sides of the equation have the same base. If the bases differ, we would need to employ other techniques, such as the change-of-base formula, to manipulate the equation into a suitable form. Additionally, we must always verify that the arguments of the logarithms are positive, as logarithms are only defined for positive numbers.

The one-to-one property significantly simplifies solving logarithmic equations. Instead of dealing with logarithms directly, we can convert the equation into a simpler algebraic equation by equating the arguments. This simplification streamlines the solution process and often leads to a straightforward algebraic manipulation to isolate the variable.

Step-by-Step Solution of $\log _2 2 x=\log _2 16$

Now, let's apply the one-to-one property to solve the given equation, $\log _2 2 x=\log _2 16$ .

Identify the Common Base: Observe that both logarithms in the equation have the same base, which is 2. This satisfies the prerequisite for applying the one-to-one property.
Apply the One-to-One Property: Since the bases are the same, we can equate the arguments of the logarithms:

$2x = 16$

This step effectively eliminates the logarithms and transforms the equation into a simple algebraic equation.
Solve for x: Now, we have a linear equation that can be solved by isolating x. Divide both sides of the equation by 2:

$x = \frac{16}{2}$

$x = 8$
Verify the Solution: It's essential to verify that the solution obtained is valid by substituting it back into the original equation and ensuring that the arguments of the logarithms remain positive. In this case, substituting $x = 8$ into the original equation gives:

$\log_2 (2 * 8) = \log_2 16$

$\log_2 16 = \log_2 16$

Since the arguments are positive and the equation holds true, the solution $x = 8$ is valid.

Therefore, the solution to the equation $\log _2 2 x=\log _2 16$ using the one-to-one property is $x = 8$ . This step-by-step solution demonstrates the power and simplicity of the one-to-one property in solving logarithmic equations.

The Importance of Verification

In the realm of solving equations, particularly those involving logarithms, the process of verification holds paramount importance. While the one-to-one property simplifies the solution process, it's crucial to remember that logarithmic functions have specific domain restrictions. Logarithms are only defined for positive arguments; therefore, any potential solution must be checked to ensure it doesn't lead to the logarithm of a non-positive number.

The act of verifying a solution involves substituting the obtained value back into the original equation. This step serves as a safeguard against extraneous solutions, which are values that satisfy the transformed equation but not the original logarithmic equation. Extraneous solutions can arise due to the nature of logarithmic transformations and the domain restrictions inherent in logarithmic functions.

In the context of our example, where we solved $\log _2 2 x=\log _2 16$ and arrived at $x = 8$ , the verification step confirms the validity of our solution. By substituting $x = 8$ back into the original equation, we obtained $\log_2 16 = \log_2 16$ , which is a true statement. Furthermore, the argument of the logarithm, $2x$ , becomes $2 * 8 = 16$ , which is positive, thus satisfying the domain requirement for logarithmic functions.

However, consider a hypothetical scenario where solving a logarithmic equation leads to a potential solution that, upon substitution, results in the logarithm of a negative number or zero. In such cases, that potential solution would be deemed extraneous and discarded. The verification step acts as a filter, separating valid solutions from extraneous ones.

Therefore, the verification process is an indispensable part of solving logarithmic equations. It not only ensures the accuracy of the solution but also reinforces a deeper understanding of the underlying principles governing logarithmic functions and their domain restrictions. Always remember to verify your solutions to guarantee their validity and avoid the pitfall of extraneous roots.

Additional Examples and Applications

The one-to-one property of logarithms is a versatile tool that can be applied to solve a wide range of logarithmic equations. Let's explore some additional examples to solidify our understanding and broaden our problem-solving skills.

Example 1: Solve $\log_3 (x + 2) = \log_3 (3x - 4)$

Identify the Common Base: The logarithms have the same base, 3.
Apply the One-to-One Property: Equate the arguments:

$x + 2 = 3x - 4$
Solve for x: Simplify the equation:

$6 = 2x$

$x = 3$
Verify the Solution: Substitute $x = 3$ back into the original equation:

$\log_3 (3 + 2) = \log_3 (3 * 3 - 4)$

$\log_3 5 = \log_3 5$

The solution is valid.

Example 2: Solve $\log (x^2 - 1) = \log 8$

Identify the Common Base: The logarithms have the same base, 10 (common logarithm).
Apply the One-to-One Property: Equate the arguments:

$x^2 - 1 = 8$
Solve for x: Simplify the equation:

$x^2 = 9$

$x = \pm 3$
Verify the Solutions: Substitute $x = 3$ and $x = -3$ back into the original equation:

For $x = 3$ :

$\log (3^2 - 1) = \log 8$

$\log 8 = \log 8$

The solution is valid.

For $x = -3$ :

$\log ((-3)^2 - 1) = \log 8$

$\log 8 = \log 8$

The solution is valid.

In this case, we obtained two solutions, both of which satisfy the original equation and the domain restrictions of the logarithmic function. These examples demonstrate the versatility of the one-to-one property in handling different types of logarithmic equations.

Conclusion

The one-to-one property of logarithms stands as a powerful tool in solving logarithmic equations. By understanding and applying this property, we can simplify complex equations into manageable algebraic forms. The key lies in recognizing equations where logarithms with the same base are equated, allowing us to equate their arguments directly. Remember, the verification step is crucial to ensure the validity of the solutions obtained, safeguarding against extraneous roots.

Through this comprehensive guide, we have explored the theoretical underpinnings of the one-to-one property, demonstrated its application in solving the equation $\log _2 2 x=\log _2 16$ , and extended our understanding through additional examples. With practice and a firm grasp of logarithmic principles, you can confidently tackle a wide array of logarithmic equations and unlock their solutions with ease. The one-to-one property is not just a technique; it's a gateway to mastering the world of logarithms and their applications in various fields of mathematics and beyond.