Math Example: Laws of Logarithms: Example 18

Topic

Logarithms

Description

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

Natural logarithms are a fundamental concept in advanced mathematics, particularly in calculus and differential equations. These examples help teach logarithmic properties by providing step-by-step demonstrations of how to manipulate and simplify logarithmic expressions involving the constant e and square roots. By working through various scenarios, students can gain a deeper understanding of the rules governing logarithms and how to apply them in different contexts.

It's crucial for students to see multiple worked-out examples to fully grasp logarithmic concepts, especially those involving natural logarithms and fractional exponents. Each example reinforces the principles while introducing slight variations, helping students recognize patterns and develop problem-solving strategies. This approach builds confidence and proficiency in handling logarithmic expressions, preparing students for more complex mathematical challenges involving exponential and logarithmic functions.

Teacher's Script: Let's look at this example involving natural logarithms and the constant e. We have ln(√(e) / e). First, we can use the quotient rule of logarithms to split this into ln(√(e)) - ln(e). Now, let's focus on ln(√(e)). Remember, √(e) is the same as e^(1/2). So, this becomes ln(e^(1/2)). Using the power rule of logarithms, this simplifies to (1/2)ln(e). What's ln(e)? It's 1. So we have 1/2 - 1. Our final answer is -1/2. This example shows how we can use logarithm properties along with the special characteristics of e to simplify complex expressions involving square roots and fractions.

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