Math Example: Laws of Logarithms: Example 31

Topic

Logarithms

Description

This example illustrates the application of the change-of-base formula to convert a quotient of two logarithms with base 4 to a natural logarithm expression. The problem involves rewriting log₄(√(e)) / log₄(e²) using natural logarithms. The solution applies the change-of-base formula log_a(x) = ln(x) / ln(a) to both the numerator and denominator, resulting in (ln(√(e)) / ln(4)) / (ln(e²) / ln(4)). This simplifies to ln(√(e)) / ln(e²), which further reduces to 1/4.

This example demonstrates the versatility of the change-of-base formula and how it can be applied to complex logarithmic expressions involving quotients. It also showcases the importance of understanding the properties of natural logarithms and the constant e. Such problems help students develop a deeper understanding of the relationships between logarithms of different bases and enhance their ability to manipulate and simplify complex logarithmic expressions.

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 examine this interesting example involving a quotient of logarithms with base 4. We start with log_4(√(e)) / log_4(e²). To convert this to natural logarithms, we'll apply the change-of-base formula to both the numerator and denominator. This gives us (ln(√(e)) / ln(4)) / (ln(e²) / ln(4)). Notice how the ln(4) terms cancel out, leaving us with ln(√(e)) / ln(e²). Now, let's simplify this further. √(e) is e^(1/2), so we have ln(e^(1/2)) / ln(e²) = (1/2) / 2 = 1/4. This example demonstrates how we can use the change-of-base formula and properties of natural logarithms to simplify complex logarithmic expressions.

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