Math Example: Laws of Logarithms: Example 32

Topic

Logarithms

Description

This example demonstrates the application of the change-of-base formula to convert a complex expression involving logarithms with base 5 to a natural logarithm expression. The problem involves rewriting (log₅(³√(e)) / log_5(e⁴)) - log₅(e) using natural logarithms. The solution applies the change-of-base formula log_a(x) = ln(x) / ln(a) to each term, resulting in (ln(³√(e)) / ln(e⁴)) - (ln(e) / ln(5)). This simplifies to (1/12 - 1) / ln(5), which further reduces to -11/12 ln(5).

This example showcases the application of multiple logarithmic properties and algebraic skills to simplify a complex logarithmic expression. It demonstrates how the change-of-base formula can be used to convert logarithms of different bases to natural logarithms, allowing for easier manipulation and simplification. Such problems help students develop a comprehensive understanding of logarithmic manipulation and enhance their problem-solving skills with advanced mathematical concepts.

By working through examples like this, students learn to approach multi-step logarithmic problems systematically, applying various logarithmic properties and algebraic skills. This type of problem-solving is crucial for success in advanced mathematics, particularly in calculus and differential equations where logarithmic and exponential functions play a significant role.

Teacher's Script: Let's tackle this complex example involving logarithms with base 5 and the constant e. We start with (log₅(³√(e)) / log₅(e⁴)) - log₅(e). To convert this to natural logarithms, we'll apply the change-of-base formula to each term. This gives us (ln(³√(e)) / ln(5)) / (ln(e⁴) / ln(5)) - ln(e) / ln(5). The ln(5) terms in the fraction cancel out, leaving us with ln(³√(e)) / ln(e⁴) - ln(e) / ln(5). Now, let's simplify further. ³√(e) is e^(1/3), so we have (ln(e^(1/3)) / ln(e⁴)) - ln(e) / ln(5) = (1/3) / 4 - 1 / ln(5) = 1/12 - 1 / ln(5). We can write this as (1/12 - 1) / ln(5) = -11/12 ln(5). This example shows how we can use the change-of-base formula and properties of logarithms to simplify complex expressions involving different bases and the constant e.

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