Math Example: Laws of Logarithms: Example 15

Topic

Logarithms

Description

This example illustrates the simplification of the natural logarithm expression ln(9e²x²) - ln(3e * x) using logarithmic properties. The solution applies the rule ln(a) - ln(b) = ln(a/b) to rewrite the expression as ln(9e²x² / 3e * x). This simplifies to ln(3e * x), which can be further expanded using the product rule of logarithms to ln(e) + ln(3x), resulting in 1 + ln(3x).

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 variables. 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 variables. 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 examine this example with natural logarithms and variables. We have ln(9e²x²) - ln(3e * x). When we subtract logarithms, we can use the quotient rule. So, this becomes ln(9e²x² / 3e * x). Now, let's simplify the fraction inside: 9e²x² divided by 3e * x is 3e * x. So we end up with ln(3e * x). Can we simplify this further? Yes, we can use the product rule of logarithms to split this into ln(3) + ln(e) + ln(x). What's ln(e)? That's right, it equals 1. So our final answer is 1 + ln(3x). This example shows how we can use logarithm properties along with the special characteristics of e to simplify complex expressions involving variables.

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