Solving Logarithmic Equations Log₅(x³) - Log₅(x²) = 2

Introduction to Logarithmic Equations

In the realm of mathematics, logarithmic equations are a fascinating and crucial area to explore. They appear in numerous scientific and engineering applications, making their understanding essential for problem-solving. Before diving into the solution for our specific equation, log₅(x³) - log₅(x²) = 2, let's first establish a solid grasp of the fundamental principles governing logarithms and their properties. At its core, a logarithm is the inverse operation to exponentiation. This means that the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. For example, the logarithm of 100 to the base 10 is 2, because 10 raised to the power of 2 equals 100. This is written mathematically as log₁₀(100) = 2. Understanding this foundational concept is vital as we delve into more complex logarithmic expressions and equations.

Logarithms come with a set of powerful properties that streamline their manipulation and simplify equations. These properties are the tools that allow us to combine, expand, and rearrange logarithmic terms, which is particularly useful when solving logarithmic equations. One of the most commonly used properties is the product rule, which states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers. Mathematically, this is expressed as logₐ(mn) = logₐ(m) + logₐ(n). Another important property is the quotient rule, which is the counterpart to the product rule and states that the logarithm of the quotient of two numbers is equal to the difference of the logarithms of the individual numbers. This is expressed as logₐ(m/n) = logₐ(m) - logₐ(n). Lastly, the power rule is essential for dealing with exponents within logarithms. It states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number, expressed as logₐ(mⁿ) = n logₐ(m). These properties, combined with the understanding of the fundamental definition of logarithms, provide the necessary foundation for tackling logarithmic equations effectively. By mastering these principles, one can confidently approach a wide array of mathematical problems involving logarithms.

Problem Statement and Initial Simplification

Now, let's focus on the specific logarithmic equation we aim to solve: log₅(x³) - log₅(x²) = 2. This equation involves logarithms with the same base, which is crucial for simplification. The base 5 logarithm, denoted as log₅, signifies the power to which 5 must be raised to obtain the argument inside the logarithm. In our equation, we have two logarithmic terms on the left-hand side, both with the same base 5, and a constant value on the right-hand side. To solve this equation, we need to consolidate the logarithmic terms and isolate the variable x. The structure of the equation, with the subtraction between the logarithmic terms, immediately suggests the application of the quotient rule of logarithms. This rule is a cornerstone in simplifying logarithmic expressions and will help us combine the two terms into a single logarithm. By applying the quotient rule, we can rewrite the equation into a more manageable form, setting the stage for further simplification and the subsequent solving process.

Applying the quotient rule of logarithms, which states that logₐ(m) - logₐ(n) = logₐ(m/n), allows us to combine the two logarithmic terms on the left-hand side of the equation. In our case, m is x³, n is x², and the base a is 5. Substituting these values into the quotient rule, we get log₅(x³) - log₅(x²) = log₅(x³/x²). This simplifies the left-hand side to a single logarithmic term, which is a significant step forward in solving the equation. Next, we can further simplify the argument inside the logarithm, which is x³/x². Using the basic rules of exponents, we know that when dividing terms with the same base, we subtract the exponents. Therefore, x³/x² simplifies to x^(3-2), which equals x¹. So, the argument inside the logarithm simplifies to x. Now, our equation looks much simpler: log₅(x) = 2. This simplified form is a major milestone in solving the equation because it directly relates the logarithm of x to the base 5 to a constant value. With this simplified equation, we are now in a position to use the definition of logarithms to solve for x. The next step involves converting this logarithmic equation into its equivalent exponential form, which will directly reveal the value of x.

Converting to Exponential Form and Solving for x

To proceed, we need to convert the simplified logarithmic equation, log₅(x) = 2, into its equivalent exponential form. The fundamental relationship between logarithms and exponentials is the key here. Recall that the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. This relationship is precisely what we need to transform our equation. In the logarithmic form log₅(x) = 2, the base is 5, the exponent is 2, and the result is x. Converting this into exponential form involves expressing x as the base raised to the exponent. Thus, we rewrite log₅(x) = 2 as 5² = x. This transformation is the heart of solving logarithmic equations, as it directly reveals the value of the variable by converting the logarithm into a simple exponential expression. By understanding and applying this conversion, we bridge the gap between logarithmic and exponential representations, paving the way for a straightforward solution.

With the equation now in exponential form, 5² = x, solving for x is a simple matter of evaluating the exponential expression. 5² means 5 raised to the power of 2, which is 5 multiplied by itself. This calculation yields 5 * 5 = 25. Therefore, we find that x = 25. This is the solution to our logarithmic equation. It's essential to understand that this value of x satisfies the original equation, making it the valid solution. However, in solving logarithmic equations, it's always a prudent step to verify the solution by plugging it back into the original equation. This verification process ensures that our solution is not extraneous, which can sometimes occur when dealing with logarithmic equations due to domain restrictions. By substituting x = 25 back into the original equation, we confirm that it satisfies the equation, thus cementing our solution. This step of verification adds a layer of certainty to our solution, making it a robust and reliable answer.

Verification and Final Solution

To ensure the accuracy of our solution, we must verify x = 25 by substituting it back into the original equation: log₅(x³) - log₅(x²) = 2. Plugging in x = 25, we get log₅(25³) - log₅(25²) = 2. Now, we need to evaluate the logarithmic terms. First, let's calculate 25³ and 25². 25³ equals 25 * 25 * 25, which is 15625. 25² equals 25 * 25, which is 625. So, our equation becomes log₅(15625) - log₅(625) = 2. Next, we need to find the logarithms of 15625 and 625 to the base 5. Recall that the logarithm is the power to which the base must be raised to obtain the number. For log₅(15625), we are looking for the power to which 5 must be raised to get 15625. Since 5⁶ = 15625, log₅(15625) = 6. Similarly, for log₅(625), we are looking for the power to which 5 must be raised to get 625. Since 5⁴ = 625, log₅(625) = 4. Substituting these values back into our equation, we get 6 - 4 = 2, which simplifies to 2 = 2. This confirms that our solution, x = 25, is indeed correct. The verification process is a crucial step in solving logarithmic equations, as it ensures that the solution is valid and satisfies the original equation without any extraneous roots.

Therefore, the final solution to the logarithmic equation log₅(x³) - log₅(x²) = 2 is x = 25. This solution represents the value that, when substituted back into the original equation, makes the equation true. Our journey to solving this equation involved several key steps, including understanding the properties of logarithms, applying the quotient rule to simplify the equation, converting the logarithmic form to exponential form, solving for x, and finally, verifying the solution. Each of these steps is an integral part of the process, and mastering them is crucial for solving a wide range of logarithmic equations. The ability to manipulate and solve logarithmic equations is not only essential in mathematics but also has applications in various fields such as physics, engineering, and computer science. Understanding the underlying principles and practicing the techniques involved will enhance problem-solving skills and provide a solid foundation for more advanced mathematical concepts.

Conclusion

In conclusion, we have successfully solved the logarithmic equation log₅(x³) - log₅(x²) = 2, and the solution we found is x = 25. This process underscored the importance of understanding and applying the fundamental properties of logarithms. We utilized the quotient rule to simplify the equation, converted the logarithmic equation into its exponential form, and solved for x. Crucially, we verified our solution to ensure its accuracy. The steps we took highlight a systematic approach that can be applied to solve other logarithmic equations. Mastery of these techniques is invaluable for anyone studying mathematics or related fields. Logarithmic equations are more than just academic exercises; they are essential tools in various scientific and engineering disciplines. The ability to confidently solve them opens doors to understanding and solving real-world problems that rely on logarithmic relationships. This exploration into logarithmic equations not only solidifies our understanding of this specific type of problem but also enhances our broader mathematical problem-solving capabilities.