Solving Logarithmic Equations Find Potential Solutions To Log₄(2-x) = Log₄(-5x-18)

In the realm of mathematics, solving equations is a fundamental skill, and logarithmic equations present a unique challenge. This article delves into the intricacies of solving the logarithmic equation:

$\log _4(2-x)=\log _4(-5 x-18)$

We will explore the steps involved in finding potential solutions and verifying their validity. This comprehensive guide will equip you with the knowledge and techniques necessary to tackle similar problems with confidence. When dealing with logarithmic equations, it's crucial to remember that the logarithm is only defined for positive arguments. This means that the expressions inside the logarithms must be greater than zero. Before we dive into the solution process, let's first understand the core concept of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simple terms, if we have an equation like $a^b = c$, then the logarithm of $c$ to the base $a$ is $b$, written as $\log_a c = b$. This means that the logarithm tells us what power we need to raise the base to in order to get a specific number. Logarithms are incredibly useful in various fields, including mathematics, physics, and engineering, for solving equations involving exponential relationships.

Understanding Logarithmic Equations

Logarithmic equations involve logarithms of expressions containing variables. The key to solving these equations lies in understanding the properties of logarithms and applying them strategically. One of the most important properties is that if $\log_a b = \log_a c$, then $b = c$, provided that $a$, $b$, and $c$ are positive and $a$ is not equal to 1. This property allows us to eliminate the logarithms and work with a simpler algebraic equation. However, it's crucial to remember that we must always check our solutions to ensure they don't lead to logarithms of negative numbers or zero, as these are undefined. In the given equation, $\log _4(2-x)=\log _4(-5 x-18)$, we have logarithms with the same base (4) on both sides. This allows us to directly apply the property mentioned above and equate the arguments of the logarithms. This step is crucial in simplifying the equation and making it easier to solve. However, before we proceed with solving the algebraic equation, let's take a moment to consider the domain of the logarithmic functions involved. The domain refers to the set of all possible input values (in this case, values of $x$) for which the function is defined. As mentioned earlier, the argument of a logarithm must be positive. Therefore, we need to ensure that both $2-x$ and $-5x-18$ are greater than zero. This will give us some initial restrictions on the possible values of $x$, which we will need to consider when we find potential solutions.

Solving the Equation

To solve the equation $\log _4(2-x)=\log _4(-5 x-18)$, we can use the property that if $\log_a b = \log_a c$, then $b = c$, provided the bases are the same and the arguments are positive. In this case, the base is 4 on both sides, so we can equate the arguments:

$2 - x = -5x - 18$

Now, we have a simple linear equation that we can solve for $x$. Let's add $5x$ to both sides:

$2 - x + 5x = -5x - 18 + 5x$

$2 + 4x = -18$

Next, subtract 2 from both sides:

$2 + 4x - 2 = -18 - 2$

$4x = -20$

Finally, divide both sides by 4:

$\frac{4x}{4} = \frac{-20}{4}$

$x = -5$

So, we have found a potential solution: $x = -5$. However, it's crucial to remember that we're dealing with logarithmic equations, and we need to verify that this solution is valid by checking if it satisfies the original equation and the domain restrictions. This is a critical step in solving logarithmic equations, as extraneous solutions can arise if we don't check our answers. Extraneous solutions are values that we obtain during the solving process but do not actually satisfy the original equation. In the context of logarithmic equations, these solutions often occur when we manipulate the equation in a way that changes the domain of the functions involved. Therefore, we must always substitute our potential solutions back into the original equation and check if they lead to logarithms of positive numbers. If a potential solution results in a logarithm of a negative number or zero, it is an extraneous solution and must be discarded. Only solutions that satisfy both the equation and the domain restrictions are considered valid.

Verifying the Solution

Now, we need to verify if $x = -5$ is a valid solution. We need to substitute this value back into the original equation and check if the arguments of the logarithms are positive:

For the first logarithm, $2 - x = 2 - (-5) = 2 + 5 = 7$. This is positive.

For the second logarithm, $-5x - 18 = -5(-5) - 18 = 25 - 18 = 7$. This is also positive.

Since both arguments are positive, we can proceed to check if the equation holds true:

$\log _4(2 - (-5)) = \log _4(7)$

$\log _4(-5(-5) - 18) = \log _4(25 - 18) = \log _4(7)$

Since $\log _4(7) = \log _4(7)$, the solution $x = -5$ is valid.

Therefore, the potential solution $x = -5$ satisfies both the equation and the domain restrictions. This confirms that it is a valid solution to the original logarithmic equation. The importance of this verification step cannot be overstated. Without it, we might mistakenly include extraneous solutions in our final answer, leading to an incorrect result. In general, when solving any equation, it's always a good practice to check your solutions by substituting them back into the original equation. This helps to ensure that you haven't made any algebraic errors and that your solutions are indeed valid. In the case of logarithmic equations, this verification step is particularly crucial due to the domain restrictions associated with logarithms.

Final Answer

Therefore, the solution to the equation $\log _4(2-x)=\log _4(-5 x-18)$ is:

$x = -5$

In conclusion, solving logarithmic equations requires a careful and methodical approach. It's essential to understand the properties of logarithms, apply them correctly, and always verify your solutions to ensure they are valid. By following these steps, you can confidently solve a wide range of logarithmic equations. This article has provided a detailed walkthrough of the solution process for a specific logarithmic equation, highlighting the key steps and considerations involved. By understanding these principles, you can apply them to solve other logarithmic equations and deepen your understanding of this important mathematical concept. Logarithmic equations appear in various applications, from calculating the magnitude of earthquakes to modeling population growth. Therefore, mastering the techniques for solving these equations is a valuable skill in many fields of study and practice.

In summary, the steps involved in solving logarithmic equations are:

1. Identify the logarithmic equation and its properties.
2. Apply the properties of logarithms to simplify the equation.
3. Solve the resulting algebraic equation.
4. Verify the solutions by substituting them back into the original equation and checking the domain restrictions.
5. Discard any extraneous solutions.
6. State the final valid solution(s).

By following these steps, you can confidently tackle logarithmic equations and arrive at accurate solutions. Remember to always pay attention to the domain restrictions and verify your answers to avoid making mistakes. With practice, you'll become more proficient in solving logarithmic equations and applying them to real-world problems. This article has aimed to provide a comprehensive guide to solving logarithmic equations, equipping you with the necessary knowledge and skills to succeed. By understanding the underlying principles and practicing regularly, you can master this important mathematical concept and apply it to various fields of study and practice.

Find the solution for the equation log₄(2-x) = log₄(-5x-18).

Solving Logarithmic Equations Find Potential Solutions to log₄(2-x) = log₄(-5x-18)