Solving The Logarithmic Equation Log Base 5 Of (10x-1) = Log Base 5 Of (9x+7)

Jul 12, 2025 by ADMIN 78 views

$Solving Logarithmic Equations A Detailed Explanation of $\log _5(10 x-1)=\log _5(9 x+7)$$

When tackling logarithmic equations, a systematic approach is essential to arrive at the correct solution. In this article, we will delve into the solution of the equation $\log _5(10 x-1)=\log _5(9 x+7)$ , providing a step-by-step explanation to enhance understanding and problem-solving skills in mathematics. This is a common type of problem encountered in algebra and precalculus courses, and mastering the techniques to solve it will be beneficial for more advanced mathematical studies.

The beauty of mathematics lies in its logical progression, where each step builds upon the previous one to reach a conclusive answer. Logarithmic equations, while sometimes appearing complex, can be simplified using the properties of logarithms. Our focus will be on clearly outlining each step, ensuring that the underlying principles are well understood. This approach will not only aid in solving the specific equation at hand but also in developing a broader understanding of how to approach similar problems.

In the sections that follow, we will first present the problem statement and then embark on a detailed journey through the solution process. This journey will involve applying logarithmic properties, algebraic manipulation, and verification of the solution. By the end of this article, you will not only know the correct answer but also understand the reasoning behind each step, empowering you to confidently solve similar equations in the future.

Understanding the Problem

The given equation is $\log _5(10 x-1)=\log _5(9 x+7)$ . This equation involves logarithms with the same base, which simplifies the solving process. The key to solving this equation lies in understanding the fundamental property of logarithms: if $\log_b(a) = \log_b(c)$ , then $a = c$ , provided that $a$ and $c$ are positive and $b$ is a positive number not equal to 1. This property allows us to eliminate the logarithms and work with a simpler algebraic equation.

Before we dive into the solution, it's crucial to recognize the domain restrictions imposed by logarithms. The argument of a logarithm must be positive. Therefore, we must ensure that both $10x - 1 > 0$ and $9x + 7 > 0$ . These inequalities will help us determine any restrictions on the possible values of $x$ and will be essential in verifying our final solution.

Ignoring these restrictions can lead to extraneous solutions, which are values that satisfy the simplified algebraic equation but not the original logarithmic equation. This is a common pitfall in solving logarithmic equations, and being mindful of the domain is a critical step in ensuring the accuracy of our solution. By carefully considering these restrictions, we can avoid errors and arrive at the correct answer with confidence.

Understanding the problem also involves recognizing the structure of the equation. We have two logarithmic expressions with the same base set equal to each other. This form suggests that we can directly equate the arguments of the logarithms, as mentioned earlier. However, before doing so, it's always a good practice to double-check that the bases are indeed the same and that the expressions are in their simplest form. This preliminary analysis can save time and prevent potential errors in the subsequent steps.

Step-by-Step Solution

Now, let's embark on the step-by-step solution of the equation $\log _5(10 x-1)=\log _5(9 x+7)$ .

Apply the Property of Logarithms: Since the bases of the logarithms are the same, we can equate the arguments:

$10x - 1 = 9x + 7$

This step is the cornerstone of solving this type of logarithmic equation. By recognizing that the logarithms have the same base, we can directly equate their arguments, transforming the logarithmic equation into a simple linear equation. This significantly simplifies the problem and allows us to apply standard algebraic techniques to find the solution.
Solve the Linear Equation: Now, we solve the linear equation for $x$ :

Subtract $9x$ from both sides:

$10x - 9x - 1 = 9x - 9x + 7$

$x - 1 = 7$

Add 1 to both sides:

$x - 1 + 1 = 7 + 1$

$x = 8$

This step involves basic algebraic manipulations to isolate $x$ . We first subtract $9x$ from both sides to group the $x$ terms together and then add 1 to both sides to isolate $x$ completely. These operations are performed with the goal of simplifying the equation and making it easier to solve for the unknown variable. The result, $x = 8$ , is a potential solution, but we must verify it against the domain restrictions before declaring it as the final answer.
Verify the Solution: We need to check if $x = 8$ satisfies the original equation and the domain restrictions. Recall that the arguments of the logarithms must be positive.

Check $10x - 1$ :

$10(8) - 1 = 80 - 1 = 79 > 0$

Check $9x + 7$ :

$9(8) + 7 = 72 + 7 = 79 > 0$

Since both arguments are positive when $x = 8$ , this solution is valid. The verification step is crucial in solving logarithmic equations because it ensures that the solution obtained algebraically is also a solution to the original logarithmic equation. By plugging the value of $x$ back into the arguments of the logarithms, we can confirm that they are positive, satisfying the domain restrictions. If the arguments were not positive, the solution would be extraneous and must be discarded.
Final Answer: The solution to the equation $\log _5(10 x-1)=\log _5(9 x+7)$ is $x = 8$ .

Having verified that $x = 8$ satisfies both the algebraic equation and the domain restrictions, we can confidently declare it as the final answer. This solution represents the value of $x$ that makes the original logarithmic equation true. The entire process, from applying the logarithmic property to verifying the solution, demonstrates a rigorous approach to solving logarithmic equations.

Common Mistakes to Avoid

When solving logarithmic equations, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and improve your problem-solving accuracy.

Ignoring Domain Restrictions: As highlighted earlier, the most common mistake is forgetting to check the domain restrictions. The argument of a logarithm must always be positive. Failing to verify this can lead to extraneous solutions. Always remember to substitute your solutions back into the original logarithmic expressions to ensure they result in positive arguments.
Incorrectly Applying Logarithmic Properties: Logarithmic properties are powerful tools for simplifying equations, but they must be applied correctly. For instance, the property $\log_b(a) = \log_b(c)$ implies $a = c$ only when the bases are the same and $a$ and $c$ are positive. Misapplying these properties can lead to erroneous simplifications and incorrect solutions. It's essential to have a solid understanding of the fundamental properties of logarithms and to use them judiciously.
Algebraic Errors: Solving logarithmic equations often involves algebraic manipulations. Making mistakes in these manipulations, such as incorrectly distributing terms or failing to combine like terms, can lead to incorrect solutions. It's crucial to be meticulous with your algebraic steps and to double-check your work to minimize the chances of errors.
Assuming Solutions Without Verification: Even if the algebraic manipulations are correct, it's essential to verify the solutions in the original equation. This step ensures that the solutions are not extraneous and that they satisfy the logarithmic equation. Skipping this step can lead to accepting incorrect answers.
Misunderstanding the Definition of Logarithms: A fundamental understanding of what a logarithm represents is crucial for solving logarithmic equations. A logarithm is the exponent to which the base must be raised to produce a given number. Misunderstanding this definition can lead to difficulties in applying logarithmic properties and solving equations. Reviewing the definition and properties of logarithms is a good way to solidify your understanding.

Practice Problems

To reinforce your understanding of solving logarithmic equations, here are a few practice problems:

$\log_2(3x + 1) = \log_2(2x + 5)$
$\log_3(x^2 - 4) = \log_3(3x)$
$\log_4(2x - 1) = 2$

Working through these problems will help you develop confidence and proficiency in solving logarithmic equations. Remember to follow the step-by-step approach outlined in this article, paying close attention to domain restrictions and verifying your solutions. The more you practice, the better you will become at recognizing patterns and applying the appropriate techniques.

Conclusion

In conclusion, solving logarithmic equations requires a systematic approach that includes understanding logarithmic properties, applying algebraic manipulations, and verifying solutions against domain restrictions. By carefully following each step and avoiding common mistakes, you can confidently solve a wide range of logarithmic equations. The solution to the equation $\log _5(10 x-1)=\log _5(9 x+7)$ is $x = 8$ . Remember to always verify your solutions to ensure they are valid. With practice and a solid understanding of the underlying principles, you can master the art of solving logarithmic equations.

The correct answer is D. x = 8.