Express $3 Log_a(4x^2) - 1/5 Log_a(8x+7)$ As A Single Logarithm

In the realm of mathematics, particularly in the study of logarithms, one often encounters expressions involving multiple logarithmic terms. The ability to condense these expressions into a single logarithm is a fundamental skill, streamlining calculations and providing a more concise representation. This article delves into the process of expressing a given logarithmic expression as a single logarithm, focusing on the application of logarithmic properties and algebraic manipulation. Let's explore how we can take a complex logarithmic expression and simplify it into a more manageable form. In this guide, we'll walk through the essential steps and properties that enable us to combine multiple logarithmic terms into one, enhancing our understanding and proficiency in handling logarithmic expressions. By mastering this technique, we can tackle a wider range of mathematical problems and appreciate the elegance and efficiency of logarithmic transformations. The simplification of logarithmic expressions is not just an academic exercise; it has practical applications in various fields such as physics, engineering, and computer science, where logarithmic scales and transformations are frequently used to model and analyze data. Understanding how to manipulate logarithms effectively is therefore a valuable asset for anyone pursuing studies or careers in these areas. Furthermore, the process of condensing logarithmic expressions reinforces our understanding of the fundamental properties of logarithms, such as the product rule, quotient rule, and power rule, which are essential for solving logarithmic equations and inequalities. By working through examples and exercises, we can develop a deeper appreciation for the interconnectedness of mathematical concepts and improve our problem-solving skills.

Problem Statement

Let's consider the following logarithmic expression that we aim to express as a single logarithm:

$$3 \log _a\left(4 x^2\right)-\frac{1}{5} \log _a(8 x+7)$$

Our objective is to use the properties of logarithms to combine these terms into a single logarithmic expression. This involves several key steps, including the application of the power rule, followed by the quotient rule, to arrive at the simplified form. The process begins with understanding the individual components of the expression and how they relate to each other within the logarithmic context. We have two logarithmic terms, each with a coefficient and an argument. The coefficients, 3 and -1/5, indicate that we will need to apply the power rule to move them as exponents of the arguments. The arguments, 4x^2 and (8x + 7), are the expressions inside the logarithms, and they will be combined using the quotient rule once the power rule is applied. The base of the logarithms, denoted as 'a', is consistent throughout the expression, which is crucial for combining the terms. If the bases were different, we would need to use the change of base formula before proceeding with the simplification. The negative sign between the two terms suggests that the quotient rule will be used, where the argument of the second term will be in the denominator of the resulting fraction. By carefully applying these principles, we can systematically transform the given expression into a single logarithm, making it easier to analyze and interpret. This simplification not only reduces the complexity of the expression but also provides a clearer understanding of the relationship between the variables and the logarithmic function.

Step-by-Step Solution

1. Apply the Power Rule

The power rule of logarithms states that $\log_b(x^p) = p \log_b(x)$. We apply this rule to both terms in the expression:

$$3 \log _a\left(4 x^2\right) = \log _a\left((4 x^2)^3\right)$$

$$-\frac{1}{5} \log _a(8 x+7) = \log _a\left((8 x+7)^{-\frac{1}{5}}\right)$$

So, our expression becomes:

$$\log _a\left((4 x^2)^3\right) + \log _a\left((8 x+7)^{-\frac{1}{5}}\right)$$

The power rule is a fundamental property of logarithms that allows us to move exponents inside the logarithm to the front as coefficients, and vice versa. This transformation is crucial for combining logarithmic terms, as it allows us to eliminate the coefficients and prepare the terms for the product or quotient rule. In this step, we apply the power rule to both terms in the expression, effectively moving the coefficients 3 and -1/5 as exponents of their respective arguments. The argument of the first term, 4x^2, is raised to the power of 3, while the argument of the second term, (8x + 7), is raised to the power of -1/5. These exponents change the form of the arguments and set the stage for the next step, where we will simplify the expressions inside the logarithms. The exponent of 3 on (4x^2) means that we will need to cube both the constant and the variable part of the argument. The exponent of -1/5 on (8x + 7) indicates that we will be taking the reciprocal of the fifth root of this expression. Understanding these implications is essential for correctly simplifying the expression in the subsequent steps. The application of the power rule is a key step in the process of expressing logarithmic expressions as a single logarithm, as it allows us to consolidate the coefficients and focus on combining the arguments.

2. Simplify the Arguments

Now, let's simplify the arguments inside the logarithms:

$$(4 x^2)^3 = 4^3 \cdot (x^2)^3 = 64 x^6$$

$$(8 x+7)^{-\frac{1}{5}} = \frac{1}{\sqrt[5]{8 x+7}}$$

Our expression now looks like this:

$$\log _a\left(64 x^6\right) + \log _a\left(\frac{1}{\sqrt[5]{8 x+7}}\right)$$

Simplifying the arguments inside the logarithms is a crucial step in the process of expressing logarithmic expressions as a single logarithm. This step involves applying algebraic rules and properties to reduce the complexity of the expressions within the logarithms. In this case, we have two arguments that need simplification: (4x²⁾3 and (8x + 7)^(-1/5). The first argument, (4x²⁾3, requires us to apply the power of a product rule, which states that (ab)^n = a^n * b^n. We raise both 4 and x^2 to the power of 3. 4^3 equals 64, and (x²⁾3 equals x^6 (using the power of a power rule, which states that (x^m)n = x^(m*n)). Therefore, (4x²⁾3 simplifies to 64x^6. The second argument, (8x + 7)^(-1/5), involves a negative fractional exponent. A negative exponent indicates that we need to take the reciprocal of the base, and a fractional exponent indicates a root. Specifically, an exponent of -1/5 means we need to take the reciprocal of the fifth root. Thus, (8x + 7)^(-1/5) simplifies to 1 / (⁵√(8x + 7)). By simplifying the arguments in this way, we prepare the logarithmic terms for the next step, which involves combining them using the product rule. The simplified arguments make the expression more manageable and easier to combine into a single logarithm. This step highlights the importance of having a strong foundation in algebraic principles when working with logarithmic expressions.

3. Apply the Product Rule

The product rule of logarithms states that $\log_b(MN) = \log_b(M) + \log_b(N)$. Applying this rule in reverse, we can combine the two logarithms into one:

$$\log _a\left(64 x^6\right) + \log _a\left(\frac{1}{\sqrt[5]{8 x+7}}\right) = \log _a\left(64 x^6 \cdot \frac{1}{\sqrt[5]{8 x+7}}\right)$$

This simplifies to:

$$\log _a\left(\frac{64 x^6}{\sqrt[5]{8 x+7}}\right)$$

The product rule of logarithms is a fundamental property that allows us to combine two logarithmic terms with the same base into a single logarithm. This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In reverse, this means that the sum of two logarithms with the same base can be expressed as a single logarithm of the product of their arguments. In this step, we apply the product rule to combine the two logarithmic terms we have: log_a(64x^6) and log_a(1 / ⁵√(8x + 7)). To do this, we multiply the arguments of the two logarithms together. This gives us (64x^6) * (1 / ⁵√(8x + 7)). Multiplying these arguments results in a single fraction where 64x^6 is the numerator and ⁵√(8x + 7) is the denominator. The expression now becomes log_a((64x^6) / ⁵√(8x + 7)). This step is crucial in our goal of expressing the original logarithmic expression as a single logarithm. By applying the product rule, we have successfully combined the two separate logarithmic terms into one, simplifying the expression and making it more concise. The resulting single logarithm represents the same value as the original expression but in a more compact form. This simplification is essential for further analysis or manipulation of the expression, and it demonstrates the power of logarithmic properties in transforming mathematical expressions.

Final Answer

Therefore, the expression $3 \log _a\left(4 x^2\right)-\frac{1}{5} \log _a(8 x+7)$ can be written as a single logarithm:

$$\log _a\left(\frac{64 x^6}{\sqrt[5]{8 x+7}}\right)$$

This final answer represents the culmination of our step-by-step simplification process, where we successfully transformed the original expression into a single logarithmic term. By applying the power rule, simplifying the arguments, and then utilizing the product rule, we were able to condense the expression into a more concise and manageable form. The final result, log_a((64x^6) / ⁵√(8x + 7)), encapsulates the original expression's value but in a significantly simplified manner. This single logarithm provides a clearer representation of the relationship between the variables and the logarithmic function, making it easier to analyze and interpret. The ability to express logarithmic expressions as a single logarithm is a valuable skill in mathematics, particularly in areas such as calculus, differential equations, and complex analysis. It allows for easier manipulation and solution of equations, as well as a deeper understanding of the underlying mathematical relationships. This process also reinforces the importance of understanding and applying the fundamental properties of logarithms, which are essential tools in mathematical problem-solving. The final answer not only provides the simplified form of the expression but also serves as a testament to the power and elegance of logarithmic transformations in mathematics.

In this article, we have demonstrated how to express a given logarithmic expression as a single logarithm. By systematically applying the power rule and the product rule of logarithms, we were able to condense the expression into a more concise form. This process not only simplifies the expression but also enhances our understanding of logarithmic properties and their applications. The ability to manipulate logarithmic expressions is a valuable skill in mathematics, with applications in various fields such as physics, engineering, and computer science. The steps we followed, from applying the power rule to simplify coefficients to using the product rule to combine terms, provide a clear roadmap for tackling similar problems. The final result showcases the elegance and efficiency of logarithmic transformations in simplifying complex mathematical expressions. This exercise also underscores the importance of a solid foundation in algebraic principles, as these are essential for manipulating the arguments within the logarithms. Furthermore, the process of expressing logarithmic expressions as a single logarithm reinforces the understanding of the inverse relationship between exponential and logarithmic functions. By mastering these techniques, students and professionals alike can enhance their problem-solving skills and gain a deeper appreciation for the power and versatility of logarithms in mathematical analysis and applications. The ability to simplify and manipulate logarithmic expressions is not just an academic exercise but a practical skill that can be applied in a wide range of real-world scenarios, from modeling physical phenomena to designing algorithms and analyzing data. Therefore, the effort invested in understanding and practicing these techniques is well worth it, as it provides a valuable tool for navigating the complexities of mathematics and its applications.