Expressing Logarithmic Expressions As A Single Logarithm

Jul 10, 2025 by ADMIN 57 views

In mathematics, especially in algebra and calculus, simplifying logarithmic expressions is a fundamental skill. Logarithms are powerful tools for manipulating and solving equations, and the ability to combine or separate logarithmic terms is essential for efficient problem-solving. This article delves into the process of expressing a given logarithmic expression, specifically ${ \log a + \frac{3}{4} \log b - 5 \log c }$ , as a single logarithm. Understanding the properties of logarithms is crucial for this task, and we will explore how these properties enable us to condense multiple logarithmic terms into one.

The expression we aim to simplify is ${ \log a + \frac{3}{4} \log b - 5 \log c }$ . To achieve this, we will utilize the properties of logarithms, which include the power rule, the product rule, and the quotient rule. The power rule allows us to move coefficients inside the logarithm as exponents, while the product and quotient rules enable us to combine logarithms of products and quotients, respectively. By applying these rules systematically, we can transform the given expression into a single logarithmic term. This process not only simplifies the expression but also provides a deeper understanding of logarithmic operations. Let’s begin by revisiting the fundamental properties of logarithms before diving into the step-by-step simplification of the given expression.

Understanding the Properties of Logarithms

Before we dive into simplifying the expression, it's crucial to understand the properties of logarithms. These properties are the foundation for manipulating logarithmic expressions and combining multiple terms into a single logarithm. There are three main properties we'll focus on: the power rule, the product rule, and the quotient rule.

1. The Power Rule

The power rule states that ${ \log_b(x^p) = p \log_b(x) }$ , where ${ b }$ is the base of the logarithm, ${ x }$ is a positive real number, and ${ p }$ is any real number. This rule allows us to move exponents inside a logarithm to the outside as coefficients, and vice versa. This is particularly useful when dealing with terms like ${ \frac{3}{4} \log b }$ or ${ 5 \log c }$ , where we need to bring the coefficients inside the logarithm to combine terms.

In our expression, we have ${ \frac{3}{4} \log b }$ and ${ 5 \log c }$ . Using the power rule, we can rewrite these terms as ${ \log(b^{\frac{3}{4}}) }$ and ${ \log(c^5) }$ , respectively. This transformation is a crucial step in combining these terms with other logarithms.

The power rule not only helps in simplifying expressions but also in solving logarithmic equations. By moving coefficients as exponents, we can often simplify equations into a form that is easier to solve. It's a fundamental property that is frequently used in various mathematical contexts involving logarithms.

2. The Product Rule

The product rule states that ${ \log_b(xy) = \log_b(x) + \log_b(y) }$ , where ${ b }$ is the base of the logarithm, and ${ x }$ and ${ y }$ are positive real numbers. This rule allows us to combine the logarithms of two numbers that are being multiplied into a single logarithm. Conversely, it also allows us to separate a single logarithm of a product into the sum of two logarithms.

In our expression, we will use this rule to combine terms that are being added. The expression ${ \log a + \log(b^{\frac{3}{4}}) }$ can be combined into ${ \log(a imes b^{\frac{3}{4}}) }$ using the product rule. This step is essential for condensing multiple logarithmic terms into a single one.

The product rule is a powerful tool in simplifying complex logarithmic expressions. It allows us to handle multiplication inside logarithms by converting it into addition outside the logarithms, making it easier to manipulate and solve equations.

3. The Quotient Rule

The quotient rule states that ${ \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) }$ , where ${ b }$ is the base of the logarithm, and ${ x }$ and ${ y }$ are positive real numbers. This rule is similar to the product rule but applies to division. It allows us to combine the logarithms of two numbers that are being divided into a single logarithm, or to separate a single logarithm of a quotient into the difference of two logarithms.

In our expression, we have a term being subtracted: ${ - 5 \log c }$ , which we rewrote as ${ - \log(c^5) }$ . When we combine this with other terms, we'll use the quotient rule. For example, if we have ${ \log(a imes b^{\frac{3}{4}}) - \log(c^5) }$ , we can rewrite this as ${ \log(\frac{a imes b^{\frac{3}{4}}}{c^5}) }$ using the quotient rule.

The quotient rule is invaluable in simplifying expressions involving division inside logarithms. It enables us to convert division into subtraction, which can be much easier to work with in many situations. This rule, along with the product and power rules, forms the cornerstone of logarithmic manipulation.

Summary of Logarithmic Properties

To recap, the key logarithmic properties we'll be using are:

Power Rule: ${ \log_b(x^p) = p \log_b(x) }$
Product Rule: ${ \log_b(xy) = \log_b(x) + \log_b(y) }$
Quotient Rule: ${ \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) }$

Understanding these properties is essential for simplifying logarithmic expressions and solving logarithmic equations. With these tools in hand, we can now proceed to simplify the given expression step-by-step.

Step-by-Step Simplification of the Expression

Now that we have a solid understanding of the properties of logarithms, we can proceed with simplifying the expression ${ \log a + \frac{3}{4} \log b - 5 \log c }$ . We will apply the power rule, product rule, and quotient rule in a systematic way to combine these logarithmic terms into a single logarithm.

Step 1: Apply the Power Rule

The first step is to apply the power rule to the terms with coefficients. Recall that the power rule states ${ \log_b(x^p) = p \log_b(x) }$ . We have two terms with coefficients: ${ \frac{3}{4} \log b }$ and ${ 5 \log c }$ . Applying the power rule, we can rewrite these as follows:

${ \frac{3}{4} \log b = \log(b^{\frac{3}{4}}) }$
${ 5 \log c = \log(c^5) }$

Now our expression looks like this:

${ \log a + \log(b^{\frac{3}{4}}) - \log(c^5) }$

This step is crucial because it eliminates the coefficients, allowing us to use the product and quotient rules to combine the logarithmic terms.

Step 2: Apply the Product Rule

The next step is to apply the product rule, which states ${ \log_b(xy) = \log_b(x) + \log_b(y) }$ . We have two terms being added: ${ \log a }$ and ${ \log(b^{\frac{3}{4}}) }$ . Applying the product rule, we combine these terms:

${ \log a + \log(b^{\frac{3}{4}}) = \log(a imes b^{\frac{3}{4}}) }$

Now our expression looks like this:

${ \log(a b^{\frac{3}{4}}) - \log(c^5) }$

We have successfully combined the first two terms into a single logarithm, making the expression simpler and closer to our goal of expressing it as a single logarithm.

Step 3: Apply the Quotient Rule

Finally, we apply the quotient rule, which states ${ \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) }$ . We have two terms being subtracted: ${ \log(a b^{\frac{3}{4}}) }$ and ${ \log(c^5) }$ . Applying the quotient rule, we can combine these terms into a single logarithm:

${ \log(a b^{\frac{3}{4}}) - \log(c^5) = \log(\frac{a b^{\frac{3}{4}}}{c^5}) }$

Thus, the simplified expression is:

${ \log(\frac{a b^{\frac{3}{4}}}{c^5}) }$

This is the final form, where the original expression has been successfully expressed as a single logarithm.

Summary of the Simplification Process

To summarize, we started with the expression ${ \log a + \frac{3}{4} \log b - 5 \log c }$ and applied the following steps:

Applied the Power Rule: Rewrote ${ \frac{3}{4} \log b }$ as ${ \log(b^{\frac{3}{4}}) }$ and ${ 5 \log c }$ as ${ \log(c^5) }$ .
Applied the Product Rule: Combined ${ \log a + \log(b^{\frac{3}{4}}) }$ into ${ \log(a b^{\frac{3}{4}}) }$ .
Applied the Quotient Rule: Combined ${ \log(a b^{\frac{3}{4}}) - \log(c^5) }$ into ${ \log(\frac{a b^{\frac{3}{4}}}{c^5}) }$ .

By systematically applying these logarithmic properties, we successfully expressed the given expression as a single logarithm: ${ \log(\frac{a b^{\frac{3}{4}}}{c^5}) }$ . This process demonstrates the power and utility of logarithmic properties in simplifying complex expressions.

Final Result and Conclusion

In conclusion, we have successfully expressed the given logarithmic expression ${ \log a + \frac{3}{4} \log b - 5 \log c }$ as a single logarithm. By applying the power rule, product rule, and quotient rule in a step-by-step manner, we have transformed the original expression into its simplified form.

The final result is:

${ \log(\frac{a b^{\frac{3}{4}}}{c^5}) }$

This single logarithmic expression is equivalent to the original sum and difference of logarithms. The ability to manipulate and simplify logarithmic expressions in this way is a crucial skill in various areas of mathematics, including algebra, calculus, and beyond. Understanding and applying these properties allows for efficient problem-solving and a deeper comprehension of logarithmic functions.

The process we followed highlights the importance of mastering logarithmic properties. Each step, from applying the power rule to combining terms using the product and quotient rules, demonstrates how these properties work together to simplify complex expressions. This simplification not only makes the expression more concise but also reveals the underlying relationships between the variables.

Furthermore, this exercise underscores the versatility of logarithms in handling multiplication, division, and exponentiation. By converting these operations into addition, subtraction, and multiplication, logarithms provide a powerful tool for solving equations and simplifying mathematical models. The ability to express multiple logarithmic terms as a single logarithm is particularly useful in solving equations where logarithms appear on both sides, as it allows for the isolation of variables and the application of inverse operations.

In summary, the simplification of ${ \log a + \frac{3}{4} \log b - 5 \log c }$ to ${ \log(\frac{a b^{\frac{3}{4}}}{c^5}) }$ is a testament to the elegance and utility of logarithmic properties. This skill is invaluable for anyone working with mathematical expressions and equations involving logarithms, and it forms a cornerstone of advanced mathematical techniques.