Solving Logarithmic Equations A Step-by-Step Guide

Jul 15, 2025 by ADMIN 51 views

This article offers a comprehensive guide on solving logarithmic equations, focusing on the specific example of $\log _5(x+2)=\log _5(8 x-5)$ . We will walk through the necessary steps, explain the underlying principles, and provide clear explanations to help you understand the process thoroughly. Whether you're a student grappling with logarithms or just looking to refresh your math skills, this guide will equip you with the knowledge to tackle similar problems with confidence. Logarithmic equations might seem daunting, but with a systematic approach, they can be solved efficiently and accurately.

Understanding Logarithmic Equations

Before diving into the solution, it's crucial to understand the basics of logarithmic equations. A logarithm is essentially the inverse operation of exponentiation. The expression $\log_b a = c$ means that $b^c = a$ , where $b$ is the base of the logarithm, $a$ is the argument, and $c$ is the exponent. When solving logarithmic equations, we're essentially trying to find the value of the variable that satisfies the equation's logarithmic relationship. The key to solving logarithmic equations lies in understanding this inverse relationship and applying the properties of logarithms correctly. A common property used is that if $\log_b a = \log_b c$ , then $a = c$ , provided that $a$ , $c$ , and $b$ meet certain conditions (specifically, $b > 0$ , $b \neq 1$ , and $a, c > 0$ ). This property allows us to eliminate the logarithms and work with a simpler algebraic equation. Another critical aspect is checking for extraneous solutions. Because the arguments of logarithms must be positive, solutions obtained algebraically might not always be valid. These extraneous solutions arise when an algebraic solution makes the argument of the logarithm negative or zero, which is undefined. Therefore, after solving the equation, it's essential to substitute the solutions back into the original equation to verify their validity. Understanding these fundamental principles ensures accurate and reliable problem-solving in the realm of logarithmic equations.

Step-by-Step Solution of $\log _5(x+2)=\log _5(8 x-5)$

Let's dive into the step-by-step solution for the equation $\log _5(x+2)=\log _5(8 x-5)$ .

Step 1: Apply the One-to-One Property

Since the bases of the logarithms are the same (base 5), we can apply the one-to-one property of logarithms. This property states that if $\log_b a = \log_b c$ , then $a = c$ . In our case, this means we can equate the arguments of the logarithms:

$x + 2 = 8x - 5$

This simplifies the equation significantly, allowing us to work with a basic algebraic equation instead of dealing with logarithms directly. The one-to-one property is a powerful tool in solving logarithmic equations because it eliminates the logarithmic functions, transforming the problem into a more manageable form. It's important to remember that this property is applicable only when the bases of the logarithms on both sides of the equation are the same. By applying this property, we pave the way for a straightforward algebraic solution, bringing us closer to finding the value(s) of $x$ that satisfy the original logarithmic equation. It's a crucial first step in many logarithmic equation problems, setting the stage for further simplification and solution finding.

Step 2: Solve the Linear Equation

Now we have a simple linear equation to solve:

$x + 2 = 8x - 5$

To solve for $x$ , we'll first move all terms involving $x$ to one side and constants to the other side. Subtract $x$ from both sides:

$2 = 7x - 5$

Next, add 5 to both sides:

$7 = 7x$

Finally, divide both sides by 7:

$x = 1$

This process of isolating $x$ is a standard algebraic technique used to solve linear equations. By carefully manipulating the equation while maintaining equality, we gradually simplify it until we arrive at the solution. In this case, the solution $x = 1$ represents the value that, if substituted into the simplified equation, would make the equation true. However, it's crucial to remember that this solution is only a potential solution to the original logarithmic equation. We must still verify this solution to ensure it doesn't lead to any undefined logarithmic expressions. This verification step is particularly important in solving logarithmic equations, as it helps us identify and discard any extraneous solutions that might arise due to the nature of logarithmic functions. Therefore, while finding $x = 1$ is a significant step, it's not the final answer until we've confirmed its validity within the context of the original logarithmic problem.

Step 3: Check for Extraneous Solutions

It is crucial to check our solution in the original logarithmic equation to ensure it doesn't lead to taking the logarithm of a non-positive number. Substitute $x = 1$ into the original equation:

$\log _5(1+2)=\log _5(8(1)-5)$ $\log _5(3)=\log _5(3)$

Since both arguments are positive (3 > 0), the solution $x = 1$ is valid. Extraneous solutions are a common pitfall in solving logarithmic equations. They arise because the domain of a logarithmic function is restricted to positive arguments. When we manipulate the equation algebraically, we might introduce solutions that don't satisfy the original domain restrictions. This is why the check for extraneous solutions is not just a formality but a critical step in the solution process. By substituting the obtained solution back into the original equation, we ensure that the arguments of all logarithms remain positive. If any argument becomes negative or zero, the solution is extraneous and must be discarded. In this specific problem, substituting $x = 1$ resulted in valid logarithmic expressions, confirming that it is indeed a legitimate solution. However, in other scenarios, this check might reveal extraneous solutions, highlighting the importance of this step in ensuring the accuracy of the final answer.

Final Answer

The solution to the logarithmic equation $\log _5(x+2)=\log _5(8 x-5)$ is:

$x = 1$

Answer: A. $x=$ 1

Common Mistakes to Avoid When Solving Logarithmic Equations

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 accuracy.

Forgetting to Check for Extraneous Solutions: This is perhaps the most common mistake. As highlighted earlier, solutions obtained algebraically must be checked in the original equation to ensure they don't result in the logarithm of a non-positive number. Failing to do so can lead to including extraneous solutions in your final answer. The argument of a logarithm must always be positive; hence, always verify that your solutions satisfy this condition. Extraneous solutions are particularly common when dealing with equations involving multiple logarithmic terms or when squaring both sides of an equation. Make it a habit to always perform this crucial check.
Incorrectly Applying Logarithmic Properties: Logarithmic properties, such as the product rule, quotient rule, and power rule, are essential tools for simplifying and solving logarithmic equations. However, misapplying these properties can lead to errors. For instance, $\log(a + b)$ is not equal to $\log a + \log b$ . Ensure you understand and correctly apply each property. A thorough understanding of logarithmic identities is crucial for manipulating logarithmic expressions accurately. Refer to the definitions and properties of logarithms whenever you're unsure, and practice applying them in different contexts to solidify your understanding.
Ignoring the Domain of Logarithmic Functions: The domain of a logarithmic function is restricted to positive numbers. Ignoring this restriction can lead to incorrect solutions. Always consider the domain of the logarithmic functions involved in the equation before and after solving. This means identifying any values of $x$ that would make the argument of any logarithm non-positive and excluding them from your solution set. Understanding and respecting the domain restrictions is fundamental to solving logarithmic equations correctly.
Incorrectly Simplifying Equations: Algebraic errors in simplifying equations are another common source of mistakes. Be careful when performing operations such as distributing, combining like terms, and isolating variables. Double-check each step to minimize the risk of errors. A systematic approach to simplification, breaking down the process into smaller, manageable steps, can help prevent mistakes. Pay close attention to signs and coefficients, and if possible, use a calculator to verify numerical computations.
Assuming the One-to-One Property Always Applies: The one-to-one property ( $\log_b a = \log_b c$ implies $a = c$ ) is a powerful tool, but it only applies when the logarithms have the same base. Attempting to apply it when the bases are different will lead to incorrect results. Before using the one-to-one property, ensure that the logarithmic expressions on both sides of the equation have the same base. If they don't, you'll need to use other techniques, such as changing the base or converting the equation into exponential form.

By being mindful of these common mistakes, you can significantly improve your ability to solve logarithmic equations accurately and efficiently.

Practice Problems

To further enhance your understanding of solving logarithmic equations, try solving these practice problems:

$\log_2(3x - 1) = 3$
$\log(x) + \log(x - 3) = 1$
$\log_4(x + 3) = \log_4(8x + 17)$

Working through these problems will help solidify your skills and build confidence in tackling various logarithmic equations. Remember to always check your solutions for extraneous solutions and to apply the logarithmic properties correctly.

Conclusion

In conclusion, solving logarithmic equations involves applying logarithmic properties, simplifying algebraic expressions, and, crucially, checking for extraneous solutions. By following the steps outlined in this guide and being aware of common mistakes, you can successfully solve a wide range of logarithmic equations. Practice regularly, and don't hesitate to review the fundamental concepts when needed. With consistent effort, you'll master the art of solving logarithmic equations and improve your overall mathematical proficiency.