Write The Expression As A Single Logarithm

Write the Expression as a Single Logarithm: A Comprehensive Guide

Condensing logarithmic expressions into a single logarithm is a fundamental skill in algebra and precalculus. Mastering this technique is crucial for solving logarithmic equations, simplifying complex expressions, and understanding the underlying properties of logarithms. This comprehensive guide will walk you through various scenarios, providing step-by-step solutions and helpful tips to solidify your understanding.

Understanding the Properties of Logarithms

Before diving into the process of combining logarithmic expressions, let's review the key properties that govern logarithmic operations. These properties are the foundation upon which all simplification techniques are built. Remember that these properties apply regardless of the base of the logarithm (unless otherwise specified).

1. Product Rule:

• Logarithm of a Product: The logarithm of a product is the sum of the logarithms of the individual factors. Mathematically, this is expressed as:

  log<sub>b</sub>(xy) = log<sub>b</sub>(x) + log<sub>b</sub>(y)

• In essence: If you're adding two logarithms with the same base, you can combine them into a single logarithm of a product.

2. Quotient Rule:

• Logarithm of a Quotient: The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. This is represented as:

  log<sub>b</sub>(x/y) = log<sub>b</sub>(x) - log<sub>b</sub>(y)

• In essence: If you're subtracting two logarithms with the same base, you can combine them into a single logarithm of a quotient.

3. Power Rule:

• Logarithm of a Power: The logarithm of a number raised to a power is the exponent multiplied by the logarithm of the base. This rule is stated as:

  log<sub>b</sub>(x<sup>p</sup>) = p * log<sub>b</sub>(x)

• In essence: If you have a coefficient in front of a logarithm, you can move it up as an exponent of the argument inside the logarithm. Conversely, you can bring an exponent down as a coefficient.

Combining Logarithmic Expressions: Examples and Explanations

Now let's tackle various examples, demonstrating how to apply these properties to write expressions as single logarithms. Each example will be broken down step-by-step to ensure clarity.

Example 1: Simple Application of the Product Rule

Write the expression log<sub>2</sub>(8) + log<sub>2</sub>(4) as a single logarithm.

Solution:

Since we are adding two logarithms with the same base (base 2), we can use the product rule:

1. Apply the Product Rule: log<sub>2</sub>(8) + log<sub>2</sub>(4) = log<sub>2</sub>(8 * 4)

2. Simplify: log<sub>2</sub>(32) = 5 (Because 2⁵ = 32)

Example 2: Simple Application of the Quotient Rule

Write the expression log<sub>10</sub>(100) - log<sub>10</sub>(10) as a single logarithm.

Solution:

We are subtracting two logarithms with the same base (base 10), so we apply the quotient rule:

1. Apply the Quotient Rule: log<sub>10</sub>(100) - log<sub>10</sub>(10) = log<sub>10</sub>(100/10)

2. Simplify: log<sub>10</sub>(10) = 1

Example 3: Combining Product and Power Rules

Write the expression 2log<sub>3</sub>(x) + log<sub>3</sub>(y) - log<sub>3</sub>(z) as a single logarithm.

Solution:

This example requires applying multiple rules in sequence:

1. Apply the Power Rule: 2log<sub>3</sub>(x) = log<sub>3</sub>(x<sup>2</sup>)

2. Apply the Product Rule: log<sub>3</sub>(x<sup>2</sup>) + log<sub>3</sub>(y) = log<sub>3</sub>(x<sup>2</sup>y)

3. Apply the Quotient Rule: log<sub>3</sub>(x<sup>2</sup>y) - log<sub>3</sub>(z) = log<sub>3</sub>(x<sup>2</sup>y/z)

Therefore, the single logarithm expression is log<sub>3</sub>(x<sup>2</sup>y/z).

Example 4: More Complex Expression

Write the expression 3log<sub>e</sub>(x) - ½log<sub>e</sub>(y) + log<sub>e</sub>(z) as a single logarithm (using natural logarithms, ln).

Solution:

Follow these steps:

1. Apply the Power Rule: 3log<sub>e</sub>(x) = log<sub>e</sub>(x<sup>3</sup>) and -½log<sub>e</sub>(y) = log<sub>e</sub>(y<sup>-1/2</sup>) = log<sub>e</sub>(1/√y)

2. Apply the Product Rule: log<sub>e</sub>(x<sup>3</sup>) + log<sub>e</sub>(1/√y) + log<sub>e</sub>(z) = log<sub>e</sub>(x<sup>3</sup> * (1/√y) * z)

3. Simplify: log<sub>e</sub>(x<sup>3</sup>z/√y)

Example 5: Dealing with Different Bases

This scenario requires a change of base formula, which is not strictly combining into a single logarithm in the same base, but it achieves simplification:

Let's say we have: log₂(8) + log₃(9)

Solution:

We cannot directly combine these because the bases are different. However, we can evaluate each individually:

log₂(8) = 3 (because 2³ = 8) log₃(9) = 2 (because 3² = 9)

Then, we can add the results: 3 + 2 = 5. This is simplified, but not a single logarithm. A true single logarithm would necessitate using the change of base formula to convert both to the same base (e.g., base 10).

Common Mistakes to Avoid

Several common mistakes can hinder your ability to correctly combine logarithmic expressions. Be mindful of these pitfalls:

• Incorrect Application of Rules: Ensure you're applying the product, quotient, and power rules accurately. Pay close attention to signs (positive or negative) and exponents.
• Mixing Bases: You can only combine logarithms that share the same base. If the bases are different, you'll need to use the change of base formula or evaluate individual logarithms separately.
• Forgetting Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). Simplify the arguments within the logarithms before applying the rules.
• Ignoring Parentheses: Parentheses are crucial for maintaining the correct order of operations and avoiding ambiguity. Use them carefully, especially when dealing with complex expressions.

Practice Problems

To solidify your understanding, try these practice problems:

1. log<sub>5</sub>(25) + log<sub>5</sub>(125)
2. log<sub>10</sub>(1000) - log<sub>10</sub>(100)
3. 2log<sub>2</sub>(x) + log<sub>2</sub>(y) - 3log<sub>2</sub>(z)
4. ½log<sub>e</sub>(x) - log<sub>e</sub>(y) + 4log<sub>e</sub>(z)
5. log₄(16) + log₂(32) - log₃(81)

Conclusion

Writing expressions as single logarithms is a valuable skill for simplifying complex mathematical expressions and solving logarithmic equations. By mastering the product, quotient, and power rules, and carefully avoiding common pitfalls, you can confidently tackle various scenarios and enhance your understanding of logarithmic functions. Remember that practice is key; the more you work through examples and problems, the more proficient you will become. Consistent practice will build your confidence and ensure mastery of this crucial algebraic concept.