Simplifying Logarithmic Expressions Rewriting As A Single Logarithm

Introduction

In this article, we will delve into the process of rewriting a complex logarithmic expression into a single, simplified logarithm. Logarithmic expressions, while powerful tools in mathematics, can often appear daunting when presented in a convoluted form. Our main goal is to simplify such expressions using the fundamental properties of logarithms. We will specifically focus on the expression: $2 ext{log} (x+3) - 3 ext{log} (x-7) + 5 ext{log} (x-2) - ext{log} (x^2)$. This expression combines several logarithmic terms with different coefficients and arguments. To rewrite this as a single logarithm, we will need to apply the power rule, the product rule, and the quotient rule of logarithms. Mastering these rules is essential for simplifying logarithmic expressions and solving related equations. Logarithms are not just abstract mathematical concepts; they have numerous applications in various fields, including physics, engineering, computer science, and finance. Understanding how to manipulate logarithmic expressions is therefore a valuable skill for anyone working in these areas. This article aims to provide a clear, step-by-step guide to simplifying the given expression, making the process accessible to anyone with a basic understanding of logarithms.

Understanding the Properties of Logarithms

To effectively rewrite the expression, it’s crucial to first understand the fundamental properties of logarithms. These properties provide the rules for manipulating logarithmic expressions and are the foundation for our simplification process. The three main properties we will use are the power rule, the product rule, and the quotient rule. The power rule states that $\log_b(a^c) = c \log_b(a)$. This rule allows us to move exponents inside the logarithm to become coefficients outside the logarithm, and vice versa. For example, $2 \log(x)$ can be rewritten as $\log(x^2)$. This property is particularly useful when dealing with terms like $2 \text{log} (x+3)$ in our given expression. Next, the product rule states that $\log_b(mn) = \log_b(m) + \log_b(n)$. This rule tells us that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In reverse, it allows us to combine the sum of logarithms into a single logarithm of the product. For instance, $\log(x) + \log(y)$ can be combined into $\log(xy)$. Lastly, the quotient rule states that $\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)$. This rule is analogous to the product rule but applies to division. It tells us that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. In reverse, it allows us to combine the difference of logarithms into a single logarithm of the quotient. For example, $\log(x) - \log(y)$ can be rewritten as $\log(\frac{x}{y})$. These three rules, the power rule, the product rule, and the quotient rule, are the cornerstones of simplifying logarithmic expressions. By understanding and applying them correctly, we can transform complex expressions into simpler, more manageable forms. In the following sections, we will apply these rules step-by-step to the given expression.

Step-by-Step Simplification Process

Now, let's apply the logarithmic properties to simplify the expression $2 \text{log} (x+3) - 3 \text{log} (x-7) + 5 \text{log} (x-2) - \text{log} (x^2)$ step by step. Our first step is to apply the power rule to each term in the expression. This means we will move the coefficients of the logarithms inside as exponents. Applying the power rule, we get: $\log ((x+3)^2) - \log ((x-7)^3) + \log ((x-2)^5) - \log (x^2)$. This transformation makes it easier to combine the terms using the product and quotient rules. Next, we will use the product rule to combine the terms that are being added. In our expression, we have $\log ((x+3)^2)$ and $\log ((x-2)^5)$ being added. Applying the product rule, we get: $\log ((x+3)^2(x-2)^5) - \log ((x-7)^3) - \log (x^2)$. This simplifies the expression by reducing the number of logarithmic terms. Now, we will apply the quotient rule to combine the terms that are being subtracted. We have two subtraction operations: subtracting $\log ((x-7)^3)$ and subtracting $\log (x^2)$. Applying the quotient rule in two steps, we first combine $\log ((x+3)^2(x-2)^5)$ and $\log ((x-7)^3)$: $\log (\frac{(x+3)^2(x-2)^5}{(x-7)^3}) - \log (x^2)$. Then, we apply the quotient rule again to combine the result with $\log (x^2)$: $\log (\frac{(x+3)^2(x-2)^5}{(x-7)^3 x^2})$. This final application of the quotient rule gives us the expression as a single logarithm. The simplified expression is $\log (\frac{(x+3)^2(x-2)^5}{(x-7)^3 x^2})$. This is the single logarithmic form of the original expression. In summary, we used the power rule to move coefficients as exponents, the product rule to combine terms being added, and the quotient rule to combine terms being subtracted. This step-by-step approach allowed us to rewrite the complex expression into a single, simplified logarithm.

Final Answer and Conclusion

After applying the power, product, and quotient rules of logarithms step-by-step, we have successfully rewritten the given expression as a single logarithm. The final simplified form of the expression $2 \text{log} (x+3) - 3 \text{log} (x-7) + 5 \text{log} (x-2) - \text{log} (x^2)$ is $\log (\frac{(x+3)^2(x-2)^5}{(x-7)^3 x^2})$. This process demonstrates the power of logarithmic properties in simplifying complex expressions. By understanding and applying these rules, we can transform multiple logarithmic terms into a single, more manageable logarithm. This skill is valuable in various mathematical contexts, including solving logarithmic equations and simplifying complex mathematical models. In conclusion, rewriting logarithmic expressions into simpler forms is a fundamental skill in mathematics. The key to success lies in a solid understanding of the properties of logarithms and a systematic, step-by-step approach. The power rule allows us to handle coefficients, while the product and quotient rules enable us to combine terms involving addition and subtraction, respectively. By mastering these techniques, you can confidently tackle a wide range of logarithmic simplification problems. This article has provided a detailed walkthrough of the simplification process, illustrating how to apply these rules effectively. Practice and familiarity with these concepts will further enhance your ability to work with logarithmic expressions.