Math Example: Laws of Logarithms: Example 26

Topic

Logarithms

Description

This example illustrates the application of the change-of-base formula to convert the sum of two logarithms with base 3 to a single logarithm with base 10. The problem involves rewriting log₃(25) + log₃(4) using the common logarithm (base 10). The solution first applies the product rule of logarithms to combine the terms into log₃(25 * 4), then uses the change-of-base formula log_a(x) = log_b(x) / log_b(a) to transform it into log(100) / log(3), which simplifies to 2 / log(3).

This example demonstrates the interplay between different logarithmic properties, including the product rule and the change-of-base formula. It showcases how these properties can be combined to simplify and transform complex logarithmic expressions. Such problems help students develop a more comprehensive understanding of logarithmic manipulation and enhance their problem-solving skills.

By working through examples like this, students learn to approach logarithmic problems strategically, choosing the most appropriate properties and techniques for each situation. This flexibility in applying logarithmic rules is essential for success in advanced mathematics and many scientific fields.

Teacher's Script: In this example, we're dealing with the sum of two logarithms with base 3, and we need to express it as a single logarithm with base 10. Let's start with log₃(25) + log₃(4). First, we can use the product rule of logarithms to combine these into a single logarithm: log₃(25 * 4) = log₃(100). Now, to change the base to 10, we apply the change-of-base formula: log₃(100) = log(100) / log(3). We know that log(100) = 2, so our final expression is 2 / log(3). This example shows how combining different logarithmic properties can help us simplify and transform expressions efficiently.

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