write the expression as a single logarithm

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29-03-2025

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This article provides a comprehensive guide on how to write expressions as single logarithms. We'll cover the key logarithmic properties and illustrate them with numerous examples, progressing from simple to more complex scenarios. Mastering this skill is crucial for success in algebra and calculus.

Understanding Logarithmic Properties

Before diving into examples, let's review the fundamental properties of logarithms that are essential for combining logarithmic expressions. These properties allow us to manipulate and simplify logarithmic equations. Remember that these properties hold true for any valid base (unless otherwise specified, we'll assume base 10 or the natural logarithm, ln, with base e).

1. Product Rule: log_b(xy) = log_b(x) + log_b(y)

This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

2. Quotient Rule: log_b(x/y) = log_b(x) - log_b(y)

The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator.

3. Power Rule: log_b(xⁿ) = n log_b(x)

The logarithm of a number raised to a power is equal to the power times the logarithm of the number.

4. Change of Base Formula: log_b(x) = log_a(x) / log_a(b)

This allows us to change the base of a logarithm from b to a. This is particularly useful when working with calculators that primarily use base 10 or base e.

Writing Expressions as Single Logarithms: Examples

Let's work through various examples, applying the properties outlined above.

Example 1: Simple Application of the Product Rule

Write the expression log₂(8) + log₂(4) as a single logarithm.

Solution: Using the product rule, we have:

log₂(8) + log₂(4) = log₂(8 * 4) = log₂(32) = 5

Example 2: Combining Product and Power Rules

Express 2log₃(x) + log₃(y) as a single logarithm.

Solution: First, apply the power rule to the first term:

2log₃(x) = log₃(x²)

Then, use the product rule:

log₃(x²) + log₃(y) = log₃(x²y)

Example 3: Using the Quotient Rule

Simplify log₅(25) - log₅(5) as a single logarithm.

Solution: Applying the quotient rule:

log₅(25) - log₅(5) = log₅(25/5) = log₅(5) = 1

Example 4: A More Complex Example

Express 3log₁₀(x) - 2log₁₀(y) + log₁₀(z) as a single logarithm.

Solution: Apply the power rule to the first two terms:

3log₁₀(x) = log₁₀(x³) -2log₁₀(y) = log₁₀(y^-2) = log₁₀(1/y²)

Now, use the product rule:

log₁₀(x³) + log₁₀(1/y²) + log₁₀(z) = log₁₀( (x³ * 1/y²) * z ) = log₁₀(x³z/y²)

Example 5: Change of Base

Convert log₂(16) to base 10.

Solution: Using the change of base formula:

log₂(16) = log₁₀(16) / log₁₀(2) ≈ 4

Troubleshooting Common Mistakes

• Incorrect order of operations: Remember to follow the order of operations (PEMDAS/BODMAS). Apply the power rule before the product or quotient rule.
• Confusing addition and subtraction: Remember that addition corresponds to multiplication (product rule) and subtraction corresponds to division (quotient rule).
• Ignoring the base: Ensure you are consistent with the base throughout your calculations.

Conclusion

Writing expressions as single logarithms is a fundamental skill in mathematics. By mastering the logarithmic properties and practicing with various examples, you can confidently tackle complex logarithmic expressions and simplify them into concise forms. Remember to always double-check your work and pay close attention to the order of operations and the base of the logarithms involved.