Math Example: Laws of Logarithms: Example 20

Topic

Logarithms

Description

This example focuses on simplifying the logarithmic expression log₂(2^{(x^2 - 1))} + log₂(2^{(x^2 + 1))} using the power property of logarithms. The solution applies the rule log_n(n^a) = a to each term, resulting in (x² - 1) + (x² + 1), which simplifies to 2x².

Logarithms with variables in the exponent are an important topic in advanced algebra and precalculus. These examples help students understand how to manipulate logarithmic expressions involving complex exponents. By working through such scenarios, students learn to apply logarithmic rules effectively and gain confidence in solving more intricate problems involving logarithms and algebraic expressions.

Providing multiple worked-out examples is crucial for students to fully grasp logarithmic concepts, especially when dealing with variable exponents. 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 logarithms involving variable exponents. We have log₂(2^{(x^2 - 1)}) + log₂(2^{(x^2 + 1)}). Remember the power rule of logarithms: when the base of the logarithm matches the base inside, we can bring down the exponent. So, log₂(2^{(x^2 - 1)}) becomes (x² - 1), and log₂(2^{(x^2 + 1)}) becomes (x² + 1). Now we add these expressions: (x² - 1) + (x² + 1) = 2x². This example demonstrates how we can simplify complex logarithmic expressions with variable exponents, a skill that's very useful in advanced algebra and calculus.

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