Math Example: Laws of Logarithms: Example 14

Topic

Logarithms

Description

This example demonstrates the simplification of the logarithmic expression log(400x³) - log(40x) using the quotient and power properties of logarithms. The solution applies the rule log(a) - log(b) = log(a/b) to rewrite the expression as log(400x³/40x). This simplifies to log(10x²), which can be further expanded using the product rule of logarithms to log(10) + log(x²), resulting in 1 + 2log(x).

Logarithms are a crucial topic in mathematics, particularly in algebra and calculus. These examples help students understand the properties of logarithms and how to manipulate logarithmic expressions involving variables. 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 variables are involved. 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 logarithms with variables. We have log(400x³) - log(40x). When we subtract logarithms, we can use the quotient rule. This means we can rewrite it as log(400x³/40x). Let's simplify the fraction inside: 400x³ divided by 40x is 10x³. So we have log(10x³). Can we simplify this further? Yes, we can use the product rule of logarithms to split this into log(10) + log(x³). What's log(10)? It's 1. And log(x³) is 2log(x). So our final answer is 1 + 2log(x). This example shows how we can use multiple logarithm properties to simplify expressions involving variables and constants.

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