Rewrite Log X + 3 Log Y - 4 Log Z As A Single Logarithm: A Comprehensive Guide

Rewriting logarithmic expressions into a single logarithm is a fundamental skill in mathematics, particularly in algebra and calculus. This process involves using the properties of logarithms to combine multiple logarithmic terms into one, making it easier to manipulate and simplify complex equations. In this comprehensive guide, we will delve into the step-by-step method of rewriting the expression log x + 3 log y - 4 log z as a single logarithm. We will explore the necessary logarithmic properties, provide detailed explanations, and offer practical examples to solidify your understanding. By mastering this technique, you will enhance your ability to solve a wide range of mathematical problems involving logarithms. This article aims to provide a clear and concise explanation of how to rewrite logarithmic expressions, ensuring that you grasp the core concepts and can apply them effectively.

Understanding Logarithmic Properties

Before we dive into rewriting the expression, it's crucial to understand the properties of logarithms that make this transformation possible. Logarithms are the inverse operations of exponentiation, and they possess several key properties that allow us to manipulate them algebraically. These properties are essential tools for simplifying and combining logarithmic expressions. Let's explore the properties that are most relevant to our task:

1. The Power Rule: The power rule states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number. Mathematically, this is expressed as:

   log_b(x^p) = p log_b(x)

   This rule is particularly useful when dealing with terms like 3 log y and 4 log z in our expression. By applying the power rule in reverse, we can rewrite these terms as single logarithms.

2. The Product Rule: The product rule states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers. Mathematically, this is expressed as:

   log_b(xy) = log_b(x) + log_b(y)

   This rule allows us to combine terms that are being added together. For example, we can combine log x and 3 log y after applying the power rule to the second term.

3. The Quotient Rule: The quotient rule states that the logarithm of the quotient of two numbers is equal to the difference of the logarithms of the individual numbers. Mathematically, this is expressed as:

   log_b(x/y) = log_b(x) - log_b(y)

   This rule is crucial for dealing with terms that are being subtracted, such as -4 log z in our expression. After applying the power rule, we can use the quotient rule to incorporate this term into a single logarithm.

These three properties—the power rule, the product rule, and the quotient rule—are the foundation for rewriting logarithmic expressions. Understanding how and when to apply these properties is key to successfully simplifying complex logarithmic equations. In the following sections, we will demonstrate how to apply these rules step-by-step to rewrite the expression log x + 3 log y - 4 log z.

Step-by-Step Rewriting of the Expression

Now that we have reviewed the fundamental properties of logarithms, let's proceed with rewriting the expression log x + 3 log y - 4 log z as a single logarithm. We will follow a systematic approach, applying the properties in a logical sequence to achieve our goal. Each step will be explained in detail to ensure clarity and understanding.

Step 1: Apply the Power Rule

The first step is to apply the power rule to the terms 3 log y and 4 log z. The power rule states that log_b(x^p) = p log_b(x). Applying this rule in reverse, we can rewrite the given terms as follows:

• 3 log y = log (y³)
• 4 log z = log (z⁴)

By applying the power rule, we have eliminated the coefficients in front of the logarithmic terms, making it easier to combine them using the product and quotient rules. The expression now becomes:

log x + log (y³) - log (z⁴)

This step is crucial as it sets the stage for the subsequent application of the product and quotient rules. Without addressing the coefficients, we cannot directly combine the logarithmic terms.

Step 2: Apply the Product Rule

Next, we apply the product rule to combine the first two terms, log x and log (y³). The product rule states that log_b(xy) = log_b(x) + log_b(y). Using this rule, we can combine these terms into a single logarithm:

log x + log (y³) = log (x * y³) = log (xy³)

Now, our expression looks like this:

log (xy³) - log (z⁴)

This step simplifies the expression further by reducing the number of logarithmic terms. We have successfully combined the terms involving addition into a single logarithm, which brings us closer to our final goal.

Step 3: Apply the Quotient Rule

Finally, we apply the quotient rule to combine the remaining terms, log (xy³) and - log (z⁴). The quotient rule states that log_b(x/y) = log_b(x) - log_b(y). Using this rule, we can rewrite the expression as:

log (xy³) - log (z⁴) = log (xy³ / z⁴)

Thus, we have successfully rewritten the original expression as a single logarithm:

log x + 3 log y - 4 log z = log (xy³ / z⁴)

This step completes the transformation, demonstrating how to combine multiple logarithmic terms into a single, simplified expression. By applying the power, product, and quotient rules in a systematic manner, we have achieved our objective.

Practical Examples and Applications

To further solidify your understanding, let's explore some practical examples and applications of rewriting logarithmic expressions. These examples will illustrate how the techniques we've discussed can be applied in various mathematical contexts.

Example 1: Simplifying Complex Logarithmic Equations

Consider the expression:

2 log a + log b - 3 log c

Following the same steps as before:

1. Apply the power rule:

   log (a²) + log b - log (c³)

2. Apply the product rule:

   log (a²b) - log (c³)

3. Apply the quotient rule:

   log (a²b / c³)

Thus, the simplified expression is log (a²b / c³). This process is crucial in solving logarithmic equations and simplifying complex mathematical expressions.

Example 2: Applications in Calculus

In calculus, logarithmic differentiation is a powerful technique used to differentiate complex functions. It often involves rewriting logarithmic expressions to simplify the differentiation process. For instance, consider a function like:

y = x² * √(x + 1) / (x - 2)³

Taking the natural logarithm of both sides gives:

ln y = ln [x² * √(x + 1) / (x - 2)³]

Rewriting the right side using logarithmic properties:

ln y = ln (x²) + ln (√(x + 1)) - ln [(x - 2)³]

ln y = 2 ln x + (1/2) ln (x + 1) - 3 ln (x - 2)

This rewritten expression is much easier to differentiate than the original. Logarithmic differentiation is widely used in calculus to handle complex functions and solve related problems.

Example 3: Applications in Physics and Engineering

Logarithms and logarithmic scales are commonly used in various fields of physics and engineering. For instance, the decibel scale, which measures sound intensity, is logarithmic. Rewriting logarithmic expressions is essential in calculations involving decibels and other logarithmic scales.

Consider the formula for sound intensity level in decibels (dB):

dB = 10 log (I / I₀)

Where:

• I is the sound intensity.
• I₀ is the reference intensity.

Rewriting and manipulating logarithmic expressions is necessary for solving problems related to sound intensity and acoustics. Similarly, in electrical engineering, logarithms are used in analyzing circuits and signals.

These examples illustrate the broad applicability of rewriting logarithmic expressions. Whether in simplifying mathematical equations, performing calculus operations, or solving real-world problems in physics and engineering, the ability to manipulate logarithms is an invaluable skill.

Common Mistakes and How to Avoid Them

While rewriting logarithmic expressions is a fundamental skill, it's also an area where mistakes can easily occur if proper care is not taken. Understanding common pitfalls and how to avoid them is crucial for ensuring accuracy in your calculations. Here are some common mistakes and strategies to prevent them:

Mistake 1: Incorrectly Applying the Power Rule

A common mistake is misapplying the power rule by either forgetting to apply it or applying it to the wrong terms. The power rule, log_b(x^p) = p log_b(x), should only be applied when an entire argument of a logarithm is raised to a power. For example, log (x²) is different from (log x)². The former can be rewritten as 2 log x, while the latter cannot be simplified using the power rule.

• How to Avoid It: Always double-check that the power applies to the entire argument inside the logarithm. Pay close attention to parentheses and exponents to ensure you are applying the rule correctly.

Mistake 2: Mixing Up the Product and Quotient Rules

Another frequent error is confusing the product and quotient rules. The product rule states that log_b(xy) = log_b(x) + log_b(y), while the quotient rule states that log_b(x/y) = log_b(x) - log_b(y). Mixing these up can lead to incorrect simplifications.

• How to Avoid It: Remember that the product rule corresponds to addition, and the quotient rule corresponds to subtraction. When terms are added, they multiply inside the logarithm; when terms are subtracted, they divide.

Mistake 3: Neglecting the Order of Operations

Like with any mathematical operation, the order of operations is crucial when rewriting logarithmic expressions. Operations inside logarithms should be simplified before applying logarithmic properties. For example, simplifying an expression like log (x² + 2x + 1) requires factoring the quadratic inside the logarithm first, if possible.

• How to Avoid It: Follow the order of operations (PEMDAS/BODMAS) diligently. Simplify inside parentheses and logarithms before applying logarithmic rules.

Mistake 4: Forgetting the Base of the Logarithm

It's essential to remember the base of the logarithm, especially when combining terms. Logarithmic properties apply only when the logarithms have the same base. If you encounter logarithms with different bases, you may need to use the change of base formula before combining them.

• How to Avoid It: Ensure that all logarithms in your expression have the same base before applying any properties. If necessary, use the change of base formula to convert them to a common base.

Mistake 5: Incorrectly Distributing Logarithms

One of the most common mistakes is attempting to distribute logarithms over addition or subtraction, which is incorrect. There is no logarithmic property that states log_b(x + y) = log_b(x) + log_b(y). This is a fundamental misunderstanding of logarithmic properties.

• How to Avoid It: Remember that logarithmic properties apply to products and quotients, not sums or differences. There is no direct way to simplify logarithms of sums or differences.

By being aware of these common mistakes and consistently applying the correct properties and order of operations, you can significantly reduce errors and improve your accuracy in rewriting logarithmic expressions.

Conclusion

In conclusion, rewriting the expression log x + 3 log y - 4 log z as a single logarithm is a valuable exercise in mastering logarithmic properties. By systematically applying the power rule, the product rule, and the quotient rule, we successfully transformed the expression into log (xy³ / z⁴). This process not only simplifies the expression but also enhances our understanding of how logarithms work.

Throughout this article, we have explored the fundamental properties of logarithms, provided a step-by-step guide to rewriting the expression, and discussed practical examples and applications in various fields, including calculus, physics, and engineering. We also highlighted common mistakes to avoid, ensuring that you can apply these techniques accurately and effectively.

Mastering the skill of rewriting logarithmic expressions is essential for anyone working with mathematical equations, scientific calculations, or engineering problems. It allows for greater flexibility in problem-solving and opens the door to more advanced mathematical concepts. By practicing these techniques and understanding the underlying principles, you can confidently tackle complex logarithmic expressions and apply them in a wide range of contexts.

Remember, the key to success in mathematics is consistent practice and a solid understanding of fundamental principles. Keep practicing, and you will find that manipulating logarithmic expressions becomes second nature. The ability to rewrite and simplify logarithms is not just a mathematical skill; it's a tool that empowers you to approach and solve a variety of real-world problems with greater confidence and precision.