Simplifying Logarithmic Expressions Log Base 5 Of (4 * 7) + Log Base 5 Of 2

In the realm of mathematics, logarithms serve as powerful tools for simplifying complex calculations and revealing underlying relationships between numbers. This article delves into the intricacies of logarithmic properties, specifically focusing on how to combine logarithmic expressions into a single, concise logarithm. We will use the expression $\log_5(4 \cdot 7) + \log_5 2$ as a practical example, demonstrating the step-by-step process of simplification and highlighting the fundamental rules governing logarithmic operations.

Logarithmic Fundamentals

Before we embark on the simplification journey, let's first establish a solid understanding of the core concepts of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, it answers the question: "To what power must we raise a base to obtain a specific number?" Mathematically, this can be expressed as:

$\log_b a = x$

where:

• $b$ represents the base of the logarithm.
• $a$ denotes the argument (the number for which we are finding the logarithm).
• $x$ is the exponent to which the base must be raised to equal the argument.

For instance, $\log_2 8 = 3$ because 2 raised to the power of 3 equals 8 (i.e., $2^3 = 8$).

Logarithms possess several key properties that enable us to manipulate and simplify expressions effectively. Among these properties, the product rule and the power rule are particularly relevant to our task at hand. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Conversely, the power rule asserts that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.

The Product Rule of Logarithms

The product rule of logarithms is a fundamental principle that allows us to simplify expressions involving the logarithm of a product. It states that the logarithm of the product of two or more numbers is equal to the sum of the logarithms of those numbers. Mathematically, this can be expressed as:

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

where:

• $b$ represents the base of the logarithm.
• $m$ and $n$ are the numbers being multiplied.

This rule is invaluable when we need to combine multiple logarithmic terms into a single logarithm or vice versa. It essentially transforms multiplication inside a logarithm into addition outside the logarithm, making it easier to manipulate and simplify expressions.

For example, consider the expression $\log_2 (8 \cdot 4)$. Using the product rule, we can rewrite this as:

$\log_2 (8 \cdot 4) = \log_2 8 + \log_2 4$

Since we know that $\log_2 8 = 3$ (because $2^3 = 8$) and $\log_2 4 = 2$ (because $2^2 = 4$), we can further simplify the expression:

$\log_2 8 + \log_2 4 = 3 + 2 = 5$

Therefore, $\log_2 (8 \cdot 4) = 5$. This demonstrates how the product rule allows us to break down a complex logarithm into simpler components, making it easier to evaluate.

The Power Rule of Logarithms

The power rule of logarithms is another essential property that helps us simplify expressions involving logarithms of numbers raised to a power. It states that the logarithm of a number raised to an exponent is equal to the product of the exponent and the logarithm of the number. Mathematically, this can be expressed as:

$\log_b (m^n) = n \log_b m$

where:

• $b$ represents the base of the logarithm.
• $m$ is the number being raised to the power.
• $n$ is the exponent.

This rule is particularly useful when dealing with expressions where the argument of the logarithm has an exponent. It allows us to move the exponent outside the logarithm as a coefficient, simplifying the expression and making it easier to work with.

For example, let's consider the expression $\log_3 (9^2)$. Using the power rule, we can rewrite this as:

$\log_3 (9^2) = 2 \log_3 9$

Since we know that $\log_3 9 = 2$ (because $3^2 = 9$), we can further simplify the expression:

$2 \log_3 9 = 2 \cdot 2 = 4$

Therefore, $\log_3 (9^2) = 4$. This example illustrates how the power rule can significantly simplify logarithmic expressions by eliminating exponents within the logarithm.

By understanding and applying the product and power rules, we can effectively manipulate logarithmic expressions, making them easier to evaluate and solve complex mathematical problems.

Applying Logarithmic Properties to Simplify the Expression

Now, let's apply these properties to simplify the expression $\log_5(4 \cdot 7) + \log_5 2$. Our goal is to combine these two logarithmic terms into a single logarithm.

Step 1: Simplify the Product Within the First Logarithm

We begin by simplifying the expression inside the first logarithm, $4 \cdot 7$, which equals 28. Thus, our expression becomes:

$\log_5(28) + \log_5 2$

Step 2: Apply the Product Rule of Logarithms

Next, we invoke the product rule of logarithms, which states that the sum of the logarithms of two numbers is equal to the logarithm of their product. In our case, this translates to:

$\log_5(28) + \log_5 2 = \log_5(28 \cdot 2)$

Step 3: Multiply the Arguments

Now, we multiply the arguments of the logarithms: 28 multiplied by 2 equals 56. Our expression now reads:

$\log_5(56)$

Therefore, the simplified form of the expression $\log_5(4 \cdot 7) + \log_5 2$ is $\log_5 56$.

Conclusion: The Power of Logarithmic Simplification

In this article, we have meticulously dissected the process of simplifying logarithmic expressions, using the example of $\log_5(4 \cdot 7) + \log_5 2$. By understanding the fundamental properties of logarithms, particularly the product rule, we successfully combined two logarithmic terms into a single, concise logarithm: $\log_5 56$. This exercise highlights the power and elegance of logarithms in simplifying complex mathematical expressions.

The ability to manipulate logarithmic expressions is crucial in various fields, including mathematics, physics, engineering, and computer science. Logarithms play a vital role in solving exponential equations, analyzing data, and modeling natural phenomena. By mastering the principles and techniques discussed in this article, you will be well-equipped to tackle a wide range of logarithmic problems and appreciate the profound impact of logarithms in the world around us.

Key Takeaways

• The product rule of logarithms is a powerful tool for combining logarithmic terms.
• Simplifying expressions within logarithms is a crucial first step.
• Understanding logarithmic properties enhances problem-solving capabilities in various fields.