Math Example: Laws of Logarithms: Example 27

Topic

Logarithms

Description

This example demonstrates the application of the change-of-base formula to convert the difference of two logarithms with base 4 to a single logarithm with base 10. The problem involves rewriting log₄(40) - log₄(4) using the common logarithm (base 10). The solution first applies the quotient rule of logarithms to combine the terms into log₄(40/4), then uses the change-of-base formula log_a(x) = log_b(x) / log_b(a) to transform it into log(10) / log(4), which simplifies to 1 / log(4).

This example highlights the importance of understanding multiple logarithmic properties and how they can be applied in sequence. It combines the quotient rule of logarithms with the change-of-base formula, showcasing how these properties work together to simplify complex expressions. Such problems help students develop a more integrated understanding of logarithmic manipulation and enhance their problem-solving skills.

By working through examples like this, students learn to approach logarithmic problems systematically, applying rules in a logical sequence to achieve the desired form. This skill is crucial for success in advanced mathematics and many scientific and engineering applications.

Teacher's Script: Let's tackle this example where we have the difference of two logarithms with base 4, and we need to express it as a single logarithm with base 10. We start with log₄(40) - log₄(4). First, we can use the quotient rule of logarithms to combine these into a single logarithm: log₄(40/4) = log₄(10). Now, to change the base to 10, we apply the change-of-base formula: log₄(10) = log(10) / log(4). We know that log(10) = 1, so our final expression is 1 / log(4). This example demonstrates how combining the quotient rule with the change-of-base formula can simplify complex logarithmic expressions efficiently.

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