Rewriting Logarithmic Expressions Using Log Properties

In the realm of mathematics, logarithms serve as a fundamental tool for simplifying complex calculations and modeling various phenomena. Logarithmic functions are the inverse of exponential functions, which means they "undo" the exponentiation operation. Understanding the properties of logarithms is essential for manipulating and simplifying logarithmic expressions effectively. In this article, we will delve into the core properties of logarithms and demonstrate how to use them to rewrite logarithmic expressions. This is a crucial skill in numerous scientific and engineering applications, where logarithmic scales are used to represent quantities that vary over a wide range, such as sound intensity (decibels), earthquake magnitudes (Richter scale), and acidity (pH scale). Mastering these properties not only aids in solving mathematical problems but also enhances the ability to interpret and analyze data in real-world contexts.

Before we dive into rewriting expressions, it's crucial to understand the fundamental properties that govern logarithmic operations. These properties allow us to manipulate logarithmic expressions, making them easier to work with and solve. The three key properties are the product rule, the quotient rule, and the power rule. Each of these rules provides a specific way to simplify or expand logarithmic expressions based on the operations within the logarithm. These properties stem directly from the corresponding properties of exponents, given the inverse relationship between logarithms and exponentials. Understanding how these rules connect to exponential properties can provide a deeper insight into why they work and how to apply them effectively.

1. Product Rule

The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is 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 rule is particularly useful when dealing with expressions involving multiplication inside the logarithm. It allows us to break down a complex logarithm into simpler terms, making it easier to evaluate or manipulate further. For example, if we have ${\log_2(8 \times 16)}$, we can rewrite it using the product rule as ${\log_2(8) + \log_2(16)}$, which simplifies to ${3 + 4 = 7}$. The product rule is a direct consequence of the exponential property that states ${b^{x+y} = b^x \times b^y}$, where the logarithm essentially undoes the exponentiation.

2. Quotient Rule

The quotient rule states that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. The formula for this rule is:

$$\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$$

where b is the base of the logarithm, and M and N are positive numbers. This rule is invaluable when dealing with division within a logarithm. It allows us to separate the logarithm of a fraction into two separate logarithmic terms, which can simplify calculations or further algebraic manipulations. For instance, the expression ${\log_3(\frac{27}{9})}$ can be rewritten using the quotient rule as ${\log_3(27) - \log_3(9)}$, which equals ${3 - 2 = 1}$. This rule is derived from the exponential property ${b^{x-y} = \frac{b^x}{b^y}}$, highlighting the close relationship between logarithmic and exponential operations.

3. Power Rule

The power rule of logarithms allows us to simplify logarithms of powers. 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. The formula for the power rule is:

$$\log_b(M^p) = p \log_b(M)$$

where b is the base of the logarithm, M is a positive number, and p is any real number. This rule is particularly useful when dealing with exponents inside a logarithm. It allows us to move the exponent outside the logarithm as a coefficient, which can greatly simplify expressions. For example, ${\log_2(4^3)}$ can be rewritten using the power rule as ${3 \log_2(4)}$, which simplifies to ${3 \times 2 = 6}$. This rule is based on the exponential property ${(b^x)^y = b^{xy}}$, further demonstrating the inverse relationship between logarithms and exponents. These rules collectively form the backbone of logarithmic manipulations, enabling us to simplify complex expressions into manageable forms.

Now, let's apply these properties to rewrite the given logarithmic expression. The expression we need to simplify is:

$$\log \frac{x^{13}}{y^4}$$

This expression involves both division and exponents, so we will use the quotient rule and the power rule to simplify it. By applying these rules systematically, we can break down the complex expression into simpler, more manageable terms. This process not only simplifies the expression but also provides a clear understanding of how the logarithmic properties work in conjunction with each other. Let's begin by addressing the division using the quotient rule.

Step 1: Apply the Quotient Rule

The first step in simplifying the given expression is to apply the quotient rule. The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms. In our case, the quotient is ${\frac{x^{13}}{y^4}}$. Applying the quotient rule, we can rewrite the expression as:

$$\log \frac{x^{13}}{y^4} = \log(x^{13}) - \log(y^4)$$

This step separates the original logarithm into two distinct logarithmic terms, each involving a power. By breaking down the expression in this way, we pave the path for further simplification using the power rule. Understanding the quotient rule is essential for handling expressions that involve division within logarithms, and it sets the stage for subsequent steps in the simplification process.

Step 2: Apply the Power Rule

Next, we apply the power rule to each term. The power rule 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. For the first term, ${\log(x^{13})}$, we apply the power rule to bring the exponent 13 down as a coefficient:

$$\log(x^{13}) = 13 \log(x)$$

Similarly, for the second term, ${\log(y^4)}$, we apply the power rule to bring the exponent 4 down as a coefficient:

$$\log(y^4) = 4 \log(y)$$

Now we have rewritten the original terms using the power rule, which simplifies the expression further by eliminating the exponents within the logarithms. This step is crucial for transforming the expression into its simplest form, making it easier to interpret and use in further calculations.

Step 3: Combine the Simplified Terms

Now that we have applied both the quotient and power rules, we can combine the simplified terms. From Step 1, we had:

$$\log \frac{x^{13}}{y^4} = \log(x^{13}) - \log(y^4)$$

From Step 2, we simplified ${\log(x^{13})}$ to ${13 \log(x)}$ and ${\log(y^4)}$ to ${4 \log(y)}$. Substituting these back into the equation, we get:

$$\log \frac{x^{13}}{y^4} = 13 \log(x) - 4 \log(y)$$

This final expression is the simplified form of the original logarithmic expression. By systematically applying the quotient and power rules, we have successfully rewritten the complex logarithm into a simpler form consisting of individual logarithmic terms with coefficients. This simplified form is much easier to work with and understand, highlighting the power and utility of logarithmic properties in simplifying mathematical expressions.

Therefore, the equivalent expression to ${\log \frac{x^{13}}{y^4}}$ using the properties of logarithms is:

$$13 \log(x) - 4 \log(y)$$

This corresponds to none of the options provided (A, B, C, or D). The correct simplification is achieved by applying the quotient rule followed by the power rule, resulting in a clear and concise expression. Understanding and applying these logarithmic properties is essential for simplifying complex expressions and solving logarithmic equations. The step-by-step process outlined above demonstrates how each property can be used systematically to transform a complicated expression into a more manageable form.

In summary, we have successfully demonstrated how to rewrite logarithmic expressions using the fundamental properties of logarithms. The key steps involved applying the quotient rule to separate the logarithm of a division into the difference of logarithms, and then using the power rule to bring down exponents as coefficients. These properties are invaluable tools for simplifying and manipulating logarithmic expressions, making them easier to work with in various mathematical and scientific contexts. Mastering these techniques not only aids in solving specific problems but also provides a deeper understanding of logarithmic functions and their applications.

By following this step-by-step approach, we transformed a complex logarithmic expression into a simpler form: ${13 \log(x) - 4 \log(y)}$. This process underscores the importance of understanding and applying the properties of logarithms accurately. While the given options did not include the correct answer, the exercise serves as a valuable illustration of how to systematically simplify logarithmic expressions. The ability to manipulate logarithms is a fundamental skill in mathematics and is crucial for tackling more advanced topics in calculus, differential equations, and various scientific fields.

By continuing to practice and apply these properties, you can enhance your understanding of logarithms and their role in mathematical problem-solving. The logarithmic properties provide a powerful set of tools for simplifying complex expressions and are essential for anyone working with mathematical models and equations.