how to write single logarithm

Mia Galupp

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03-04-2025

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Are you struggling to combine multiple logarithms into a single, more concise expression? Mastering this skill is crucial for simplifying complex logarithmic equations and solving problems in algebra, calculus, and beyond. This guide will walk you through the process, providing clear explanations and examples to help you confidently condense logarithmic expressions into a single logarithm.

Understanding the Properties of Logarithms

Before we dive into condensing, let's review the fundamental properties of logarithms that govern these manipulations. These properties are essential for understanding how and why we can combine logarithms.

• Product Rule: log_b(xy) = log_b(x) + log_b(y) This rule states that the logarithm of a product is the sum of the logarithms.

• Quotient Rule: log_b(x/y) = log_b(x) - log_b(y) This rule shows that the logarithm of a quotient is the difference of the logarithms.

• Power Rule: log_b(xⁿ) = n log_b(x) This rule states that the logarithm of a number raised to a power is the exponent multiplied by the logarithm of the number.

Important Note: These rules apply only when the logarithms share the same base (represented by 'b').

Condensing Logarithmic Expressions: Step-by-Step Guide

Now, let's apply these rules to condense multiple logarithms into a single logarithm. We'll work through several examples to illustrate the process.

Example 1: Using the Product Rule

Let's say we have the expression: log₂(8) + log₂(4).

1. Identify the common base: Both logarithms have a base of 2.
2. Apply the Product Rule: Since we have a sum of logarithms with the same base, we can use the product rule: log₂(8) + log₂(4) = log₂(8 * 4) = log₂(32).

Therefore, the condensed form is log₂(32).

Example 2: Using the Quotient Rule

Consider the expression: log₁₀(100) - log₁₀(10).

1. Identify the common base: Both logarithms have a base of 10.
2. Apply the Quotient Rule: Since we're subtracting logarithms with the same base, we apply the quotient rule: log₁₀(100) - log₁₀(10) = log₁₀(100/10) = log₁₀(10).

Therefore, the condensed form is log₁₀(10), which simplifies further to 1 (because 10¹ = 10).

Example 3: Combining Product and Power Rules

Let's tackle a more complex example: 3log₃(x) + log₃(y) - log₃(z).

1. Apply the Power Rule: First, address the coefficient '3' in front of log₃(x) using the power rule: 3log₃(x) = log₃(x³).
2. Apply the Product and Quotient Rules: Now, combine the terms using the product and quotient rules: log₃(x³) + log₃(y) - log₃(z) = log₃((x³y)/z).

Therefore, the condensed form is log₃((x³y)/z).

Practice Makes Perfect!

Condense the following logarithmic expressions as practice:

1. log₅(25) + log₅(5)
2. log_e(x²) - log_e(y)
3. 2log₁₀(a) + log₁₀(b) - log₁₀(c)

By consistently practicing these techniques, you’ll build a strong understanding of how to effectively condense logarithmic expressions. Remember to always check that the logarithms share the same base before applying the rules! This skill will serve you well in your mathematical endeavors.