Math Example: Laws of Logarithms: Example 16

Topic

Logarithms

Description

This example demonstrates the simplification of the logarithmic expression log₂(√(2)/2) using logarithmic properties. The solution applies the rule log_n(a/b) = log_n(a) - log_n(b) to rewrite the expression as log_2(√(2)) - log_2(2). This simplifies to (1/2)log₂(2^1/2) - 1, which further reduces to 1/2 - 1, resulting in -1/2.

Logarithms with different bases are a crucial topic in mathematics, particularly in algebra and precalculus. These examples help students understand the properties of logarithms and how to manipulate logarithmic expressions involving square roots and fractions. By working through various scenarios, students learn to apply logarithmic rules effectively and gain confidence in solving more complex problems involving logarithms and algebraic expressions.

Providing multiple worked-out examples is essential for students to fully grasp logarithmic concepts, especially when dealing with different bases and fractional exponents. Each example reinforces the basic principles while introducing slight variations, helping students recognize patterns and develop problem-solving strategies. This approach allows students to see how logarithmic properties can be applied in different contexts, enhancing their understanding and ability to tackle diverse logarithmic problems in algebra and other mathematical fields.

Teacher's Script: Now, let's look at this example involving a logarithm with base 2. We have log₂(√(2)/2). First, we can use the quotient rule of logarithms to split this into log₂(√(2)) - log₂(2). Now, let's focus on log₂(√(2)). Remember, √(2) is the same as 2^(1/2). So, this becomes (1/2)log₂(2). What's log₂(2)? It's 1. So we have 1/2 - 1. Our final answer is -1/2. This example shows how we can use logarithm properties to simplify expressions involving square roots and fractions, even when working with a base other than 10 or e.

For a complete collection of math examples related to Logarithms click on this link: Math Examples: Laws of Logarithms Collection.