Math Example: Laws of Logarithms: Example 24

Topic

Logarithms

Description

This example focuses on simplifying the natural logarithm expression ln(√(e)/(x - 1)) + ln(x - 1). The solution applies the quotient property of logarithms to split ln(√(e)/(x - 1)) into ln(√(e)) - ln(x - 1). Then, it uses the property ln(√(e)) = ln(e^(1/2)) = 1/2 * ln(e) = 1/2. The ln(x - 1) terms cancel out, leaving the final result as 1/2.

Natural logarithms involving algebraic fractions and roots are an important topic in advanced calculus and differential equations. These examples help students understand how to manipulate complex logarithmic expressions involving variables and radicals. By working through such scenarios, students learn to apply logarithmic rules effectively and gain confidence in solving more intricate problems involving exponential and logarithmic functions.

Providing multiple worked-out examples is crucial for students to fully grasp natural logarithm concepts, especially when dealing with algebraic fractions and roots. Each example reinforces the basic principles while introducing more complex variations, helping students recognize patterns and develop problem-solving strategies for a wide range of logarithmic expressions in calculus and related fields.

Teacher's Script: Let's examine this example with a natural logarithm of a fraction involving e and a variable. We need to simplify ln(√(e)/(x - 1)) + ln(x - 1). First, let's use the quotient property of logarithms to split the fraction: ln(√(e)) - ln(x - 1) + ln(x - 1). Notice that the ln(x - 1) terms will cancel out. Now, let's focus on ln(√(e)). We can rewrite this as ln(e^(1/2)), which equals (1/2)ln(e). Since ln(e) = 1, our final result is simply 1/2. This example demonstrates how we can simplify complex natural logarithm expressions by carefully applying logarithm properties and recognizing terms that cancel out.

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