Simplifying Logarithmic Expressions A Comprehensive Guide

In the realm of mathematics, logarithmic expressions often appear complex and daunting. However, with a systematic approach and a solid understanding of logarithmic properties, these expressions can be simplified effectively. This article delves into the process of simplifying a specific logarithmic expression, providing a step-by-step guide and explanations to enhance comprehension. By mastering these techniques, you can confidently tackle various logarithmic challenges.

Understanding Logarithms

Before diving into the simplification process, it’s crucial to grasp the fundamental concepts of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, if we have an exponential equation like b^x = y, the logarithmic form is expressed as log_b y = x. Here, b is the base of the logarithm, y is the argument, and x is the exponent. Logarithms help us determine the exponent to which a base must be raised to obtain a specific value.

Logarithms possess several key properties that are instrumental in simplifying expressions. These properties include:

1. Product Rule: log_b (mn) = log_b m + log_b n
2. Quotient Rule: log_b (m/n) = log_b m - log_b n
3. Power Rule: log_b (m^p) = p log_b m
4. Change of Base Formula: log_b a = log_c a / log_c b

These rules allow us to manipulate logarithmic expressions, combining or separating terms to facilitate simplification. Understanding these rules is paramount to successfully navigating logarithmic problems.

Problem Statement

Let's consider the logarithmic expression we aim to simplify:

(2 + 3 log₂ 9 + 4 log₂ 3 - log₂ 27) / (3 + log₂ 81)

This expression involves multiple logarithmic terms with the same base, which makes it amenable to simplification using the properties we discussed earlier. The key is to strategically apply these rules to reduce the complexity of the expression.

Step-by-Step Simplification

To simplify the given expression, we will proceed step by step, applying the properties of logarithms meticulously. Each step will be explained in detail to ensure clarity.

Step 1: Rewrite the terms using the power rule.

The power rule of logarithms states that log_b (m^p) = p log_b m. We can use this rule to rewrite the terms involving coefficients:

• 3 log₂ 9 can be rewritten as log₂ (9³)
• 4 log₂ 3 can be rewritten as log₂ (3⁴)
• log₂ 27 can be rewritten as log₂ (3³)
• log₂ 81 can be rewritten as log₂ (3⁴)

Substituting these back into the original expression, we get:

(2 + log₂ (9³) + log₂ (3⁴) - log₂ (3³)) / (3 + log₂ (3⁴))

Step 2: Simplify the arguments.

Now, let's simplify the arguments of the logarithms:

• 9³ = (3²)³ = 3⁶
• 3⁴ = 81
• 3³ = 27

Our expression now becomes:

(2 + log₂ (3⁶) + log₂ (81) - log₂ (27)) / (3 + log₂ (81))

Step 3: Rewrite constants as logarithms.

To combine the constant terms with the logarithmic terms, we need to express the constants as logarithms with base 2. Recall that log₂ 2 = 1. Therefore:

• 2 can be rewritten as 2 log₂ 2 = log₂ (2²)
• 3 can be rewritten as 3 log₂ 2 = log₂ (2³)

Substituting these into the expression, we have:

(log₂ (2²) + log₂ (3⁶) + log₂ (81) - log₂ (27)) / (log₂ (2³) + log₂ (81))

Step 4: Apply the product and quotient rules.

The product rule states that log_b (mn) = log_b m + log_b n, and the quotient rule states that log_b (m/n) = log_b m - log_b n. We will use these rules to combine the logarithmic terms in the numerator and the denominator.

Numerator:

log₂ (2²) + log₂ (3⁶) + log₂ (81) - log₂ (27) = log₂ [(2² * 3⁶ * 81) / 27]

Denominator:

log₂ (2³) + log₂ (81) = log₂ (2³ * 81)

Our expression now looks like this:

log₂ [(2² * 3⁶ * 81) / 27] / log₂ (2³ * 81)

Step 5: Simplify the arguments further.

Let’s simplify the arguments inside the logarithms:

Numerator Argument:

(2² * 3⁶ * 81) / 27 = (4 * 729 * 81) / 27 = (4 * 729 * 3³) / 3³ = 4 * 729 = 4 * 3⁶ = 2916

Denominator Argument:

2³ * 81 = 8 * 81 = 648

So, the expression becomes:

log₂ (2916) / log₂ (648)

Step 6: Prime factorization.

To further simplify, we express the arguments in terms of their prime factors:

• 2916 = 2² * 3⁶
• 648 = 2³ * 3⁴

The expression now is:

log₂ (2² * 3⁶) / log₂ (2³ * 3⁴)

Step 7: Apply logarithm properties again.

Using the product rule, we separate the logarithms:

Numerator:

log₂ (2² * 3⁶) = log₂ (2²) + log₂ (3⁶)

Denominator:

log₂ (2³ * 3⁴) = log₂ (2³) + log₂ (3⁴)

Now, using the power rule, we get:

(2 log₂ 2 + 6 log₂ 3) / (3 log₂ 2 + 4 log₂ 3)

Since log₂ 2 = 1, the expression simplifies to:

(2 + 6 log₂ 3) / (3 + 4 log₂ 3)

Step 8: Evaluate Final Answer

We will solve it using substitution method. Let $log_2 3 = x$:

(2 + 6x) / (3 + 4x)

This is the simplified form of the original expression. Without additional context or instructions, this can be considered the final simplified form.

Conclusion

Simplifying logarithmic expressions requires a strong foundation in logarithmic properties and a systematic approach. By applying the product, quotient, and power rules, along with strategic manipulation, complex expressions can be reduced to simpler forms. In this article, we meticulously simplified the expression (2 + 3 log₂ 9 + 4 log₂ 3 - log₂ 27) / (3 + log₂ 81), demonstrating each step in detail. Mastering these techniques will empower you to confidently tackle a wide range of logarithmic problems.

Understanding logarithms and their properties is not just a mathematical exercise; it has practical applications in various fields such as computer science, physics, and engineering. The ability to simplify logarithmic expressions is a valuable skill for anyone working with exponential relationships and mathematical models.

Through practice and a clear understanding of the fundamental rules, simplifying logarithmic expressions becomes a manageable and even enjoyable mathematical endeavor. Remember to break down complex problems into smaller, more manageable steps, and always double-check your work to ensure accuracy.