Math Example: Laws of Logarithms: Example 22

Topic

Logarithms

Description

This example focuses on simplifying the logarithmic expression log(√(10)/(x + 1)) + log(x + 1). The solution applies the quotient property of logarithms to split log(√(10)/(x + 1)) into log(√(10)) - log(x + 1). Then, it uses the property log(√(10)) = log(10^(1/2)) = (1/2)log(10) = 1/2. The log(x + 1) terms cancel out, leaving the final result as 1/2.

Logarithms involving algebraic fractions and roots are an important topic in advanced algebra and precalculus. 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.

Providing multiple worked-out examples is crucial for students to fully grasp logarithmic 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.

Teacher's Script: Let's examine this example with a logarithm of a fraction and a square root. We need to simplify log(√(10)/(x + 1)) + log(x + 1). First, let's use the quotient property of logarithms to split the fraction: log(√(10)) - log(x + 1) + log(x + 1). Notice that the log(x + 1) terms will cancel out. Now, let's focus on log(√(10)). We can rewrite this as log(10^(1/2)), which equals (1/2)log(10). Since log(10) = 1, our final result is simply 1/2. This example demonstrates how we can simplify complex logarithmic 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.