Expanding Logarithmic Expressions: A Step-by-Step Guide

Nov 24, 2025 by Alex Johnson 56 views

Understanding and manipulating logarithmic expressions is a fundamental skill in mathematics. Logarithms, at first glance, might seem complex, but they are powerful tools for simplifying calculations and solving equations, especially those involving exponents. This article will guide you through the process of expanding logarithmic expressions using the properties of logarithms, focusing on an example that involves radicals and multiple variables. By mastering these techniques, you'll be able to tackle more advanced mathematical problems with confidence. We'll break down the process into manageable steps, ensuring clarity and comprehension along the way.

Understanding the Properties of Logarithms

Before diving into the expansion of a specific logarithmic expression, it's crucial to grasp the fundamental properties of logarithms. These properties serve as the bedrock for manipulating and simplifying logarithmic expressions. Let's explore these key principles that govern how logarithms behave:

Product Rule: 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). This rule allows us to break down complex logarithmic expressions involving multiplication into simpler terms. For example, if you have log(2x), you can expand it to log(2) + log(x). This is particularly useful when dealing with expressions inside the logarithm that can be separated into multiple components.
Quotient Rule: The logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. In mathematical notation, this is written as log_b(M/N) = log_b(M) - log_b(N). This rule is invaluable for simplifying expressions where the argument of the logarithm is a fraction. Imagine you have log(x/y); you can rewrite it as log(x) - log(y). This transformation simplifies the expression and makes it easier to work with, especially in solving equations.
Power Rule: The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. This is represented as log_b(M^p) = p * log_b(M). The power rule is exceptionally useful when dealing with exponents within logarithms. If you encounter log(x²), you can use this rule to bring the exponent down and rewrite it as 2 * log(x). This greatly simplifies the expression and is a cornerstone for solving logarithmic equations.
Change of Base Rule: This rule allows you to convert logarithms from one base to another. It's particularly useful when dealing with logarithms that do not have a common base (like base 10 or base e). The change of base rule is expressed as log_b(a) = log_c(a) / log_c(b), where c is the new base. This rule is essential when you need to evaluate logarithms on a calculator that only supports certain bases. For example, if you need to find log₅(10), you can use the change of base rule to convert it to log₁₀(10) / log₁₀(5), which can be easily calculated.

Understanding and skillfully applying these properties of logarithms is essential for simplifying complex expressions and solving logarithmic equations. Each rule serves a specific purpose, and knowing when and how to apply them is key to mastering logarithmic manipulations. With these rules in your arsenal, you'll be well-equipped to tackle a wide array of logarithmic problems.

Expanding the Expression: log(³√(x⁷y²z))

Now, let's apply these properties to expand the given logarithmic expression: log(³√(x⁷y²z)). This expression involves a cube root, multiple variables, and exponents, making it a perfect example to illustrate the power of logarithmic properties. We will break down the expansion process step-by-step to ensure clarity and understanding.

Step 1: Convert the Radical to an Exponent

The first step in expanding the expression is to convert the cube root into its equivalent exponential form. Recall that the nth root of a number can be written as that number raised to the power of 1/n. Therefore, the cube root of (x⁷y²z) can be expressed as (x⁷y²z)^(1/3). Substituting this into the original expression, we get:

log(³√(x⁷y²z)) = log((x⁷y²z)^(1/3))

This transformation is crucial because it allows us to apply the power rule of logarithms in the next step. By converting the radical to an exponent, we make the expression more amenable to logarithmic manipulation. This is a common technique in simplifying expressions involving roots and logarithms, and it's a skill that's widely applicable in various mathematical contexts.

Step 2: Apply the Power Rule

The next step involves applying the power rule of logarithms. As we learned earlier, the power rule states that log_b(M^p) = p * log_b(M). In our case, M is (x⁷y²z) and p is 1/3. Applying the power rule, we bring the exponent 1/3 down as a coefficient:

log((x⁷y²z)^(1/3)) = (1/3) * log(x⁷y²z)

This step significantly simplifies the expression by removing the exponent from within the logarithm. The coefficient (1/3) now multiplies the entire logarithmic expression, which will be further expanded in the subsequent steps. Recognizing and applying the power rule is a key step in simplifying logarithmic expressions, particularly when dealing with exponents or radicals.

Step 3: Apply the Product Rule

Now, we have the expression (1/3) * log(x⁷y²z). Inside the logarithm, we have a product of three terms: x⁷, y², and z. The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Therefore, we can expand log(x⁷y²z) as follows:

log(x⁷y²z) = log(x⁷) + log(y²) + log(z)

Substituting this back into our expression, we get:

(1/3) * log(x⁷y²z) = (1/3) * [log(x⁷) + log(y²) + log(z)]

This step breaks down the single logarithm into a sum of three separate logarithms, each involving one variable. This is a significant simplification, as it isolates each variable and prepares the expression for further expansion using the power rule again.

Step 4: Apply the Power Rule Again

We've now reached a point where we have logarithms with exponents inside them: log(x⁷) and log(y²). We can apply the power rule once more to eliminate these exponents. Applying the power rule to log(x⁷) and log(y²), we get:

log(x⁷) = 7 * log(x) log(y²) = 2 * log(y)

Substituting these back into our expression, we have:

(1/3) * [log(x⁷) + log(y²) + log(z)] = (1/3) * [7log(x) + 2log(y) + log(z)]

This step is crucial for achieving the final expanded form, where each logarithm involves only one variable and has no exponents or radicals. By applying the power rule again, we've successfully brought down the exponents as coefficients, further simplifying the expression.

Step 5: Distribute the Coefficient

The final step is to distribute the coefficient (1/3) across the terms inside the brackets:

(1/3) * [7log(x) + 2log(y) + log(z)] = (7/3)log(x) + (2/3)log(y) + (1/3)log(z)

This step completes the expansion of the logarithmic expression. We have successfully expanded the original expression into a sum of individual logarithms, each involving only one variable and with no radicals or exponents. This is the final form of the expanded logarithmic expression.

Final Expanded Expression

Therefore, the fully expanded form of the given logarithmic expression log(³√(x⁷y²z)) is:

(7/3)log(x) + (2/3)log(y) + (1/3)log(z)

This expanded form clearly demonstrates the application of the properties of logarithms, breaking down a complex expression into simpler, manageable terms. Each step in the process utilizes a specific property, showcasing the power and versatility of logarithms in mathematical manipulations. By following these steps, you can confidently expand a wide range of logarithmic expressions.

Conclusion

Expanding logarithmic expressions using the properties of logarithms is a fundamental skill in mathematics. By understanding and applying the product rule, quotient rule, and power rule, you can simplify complex expressions into more manageable forms. In this article, we took the expression log(³√(x⁷y²z)) and systematically expanded it step-by-step, demonstrating how to convert radicals to exponents, apply the power rule, and break down products into sums of logarithms. The final expanded form, (7/3)log(x) + (2/3)log(y) + (1/3)log(z), showcases the power of these properties in simplifying mathematical expressions. Mastering these techniques will enable you to tackle more advanced mathematical problems with greater confidence and proficiency.

For further exploration and practice on logarithmic properties, you can visit Khan Academy's Logarithm section, a trusted resource for mathematics education.