$\log _5(4 \cdot 7)+\log _5 2$ As A Single Logarithm Explained

Understanding Logarithmic Properties

To solve this problem, we need to understand the fundamental properties of logarithms. Specifically, we will use the product rule of logarithms, 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 can be expressed as:

$\log_b(mn) = \log_b(m) + \log_b(n)$

where $b$ is the base of the logarithm, and $m$ and $n$ are positive numbers. This property is crucial for combining multiple logarithmic terms into a single logarithm, which is what the question asks us to do. In essence, the product rule allows us to reverse the process of expanding a logarithm of a product into separate logarithms. When we encounter an expression like $\log_5(4 \cdot 7)+\log_5 2$, the product rule provides the mechanism to simplify and consolidate it effectively. Understanding and applying this rule correctly is paramount for solving logarithmic equations and simplifying expressions in various mathematical contexts. The ability to manipulate logarithmic expressions using such rules demonstrates a strong grasp of logarithmic functions and their properties, which is vital in fields ranging from mathematics and physics to computer science and engineering.

Furthermore, another essential logarithmic property that we will utilize is the ability to simplify the logarithm of a product within the argument of the logarithmic function. In the given expression, we have $\log_5(4 \cdot 7)$, which represents the logarithm base 5 of the product of 4 and 7. Before applying the sum-to-product rule, it's often beneficial to simplify this product. This means we first calculate $4 \cdot 7 = 28$, which transforms the expression into $\log_5(28)$. Simplifying the argument in this way can make subsequent steps clearer and prevent potential errors in calculation. This step exemplifies a broader principle in mathematics: simplification before proceeding with more complex operations. By reducing expressions to their simplest forms early on, we often gain clarity and ease the application of more advanced rules or theorems. This practice not only enhances accuracy but also deepens our understanding of the underlying mathematical structures. In the context of logarithmic functions, simplifying the argument before combining terms or applying other logarithmic properties is a hallmark of efficient and precise problem-solving.

Applying the Product Rule

In our problem, we have the expression $\log _5(4 \cdot 7)+\log _5 2$. The first step is to simplify the expression inside the first logarithm. We have:

$4 \cdot 7 = 28$

So, our expression becomes:

$\log _5(28) + \log _5 2$

Now, we can apply the product rule of logarithms. According to the product rule, the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments. In this case, the base is 5, and the arguments are 28 and 2. Therefore, we can rewrite the expression as:

$\log _5(28 \cdot 2)$

This step is a direct application of the logarithmic product rule, showcasing how it simplifies expressions involving the addition of logarithms. The ability to transition smoothly from a sum of logarithms to a single logarithm of a product is a fundamental skill in logarithmic manipulations. It demonstrates a clear understanding of the inverse relationship between logarithmic and exponential functions. By correctly applying this rule, we not only simplify the given expression but also pave the way for further calculations or simplifications, if necessary. The transformation $\log _5(28) + \log _5 2$ to $\log _5(28 \cdot 2)$ is a testament to the power and elegance of logarithmic properties in mathematical problem-solving. This step is not just a mechanical application of a rule but an insightful utilization of a mathematical principle to condense and clarify an expression.

Simplifying the Result

Next, we need to multiply the numbers inside the logarithm:

$28 \cdot 2 = 56$

So, our expression simplifies to:

$\log _5(56)$

This is a single logarithm, as the question requested. Thus, the expression $\log _5(4 \cdot 7)+\log _5 2$ written as a single log is $\log _5(56)$. This final simplification step underscores the importance of performing basic arithmetic operations within the logarithmic expression to arrive at the most concise form. Multiplying 28 by 2 to obtain 56 is a straightforward yet crucial step in completing the simplification process. It transforms the expression from a product within the logarithm to a single numerical argument, making it easier to interpret and utilize in further calculations if needed. This step also highlights the interconnectedness of different mathematical concepts, where basic arithmetic skills are essential even in the context of more advanced topics like logarithms. The transition from $\log _5(28 \cdot 2)$ to $\log _5(56)$ demonstrates how careful attention to detail in each step of a mathematical problem leads to an accurate and simplified result. The final expression, $\log _5(56)$, is not only a single logarithm but also represents the most streamlined form of the original expression, adhering to the problem's requirement of writing it as a single log.

Conclusion

Therefore, the expression $\log _5(4 \cdot 7)+\log _5 2$ written as a single log is:

$\log _5(56)$

So, the correct answer is:

D. $\log _5 56$

This problem demonstrates the application of the product rule of logarithms and the importance of simplifying expressions step by step. The solution showcases how understanding and correctly applying logarithmic properties can effectively consolidate logarithmic terms. By breaking down the problem into smaller, manageable steps, we first simplified the initial expression, then applied the product rule to combine the logarithms, and finally, performed the necessary arithmetic to arrive at the single logarithmic expression. The final answer, $\log _5(56)$, is the culmination of these steps, reflecting the power of logarithmic rules in simplifying mathematical expressions. This exercise is a clear illustration of how mathematical concepts build upon each other, with basic arithmetic being crucial for handling more advanced topics like logarithms. The ability to manipulate logarithmic expressions accurately is a valuable skill in various scientific and engineering disciplines, making this type of problem a fundamental component of mathematical education.