# Lesson 18

Applications of Logarithmic Functions

### Lesson Narrative

This lesson is optional because it goes beyond the depth of understanding required to address the standards. Students encounter real-world applications of logarithmic functions and use them to solve problems. They build log functions to represent the acidity of substances and the intensity of earthquakes. These applications give students opportunities to interpret key features of tables containing real-life data that can be modeled mathematically and to write functions to describe the relationships.

To describe and generalize the patterns in this lesson, students need to make use of structure (MP7). The quantitative relationships here may also be unfamiliar to students, so making sense of them requires persistence (MP1).

Students are not formally introduced to a logarithmic scale in this lesson.

### Learning Goals

Teacher Facing

• Use logarithmic relationships to solve problems about acidity and earthquakes.

### Student Facing

• Let’s measure acidity levels and earthquake strengths.

### Required Preparation

If possible, obtain pH testing strips for the activity How Acidic Is It?

### Student Facing

• I understand how logarithms are used to measure things like acidity and the intensity of earthquakes.

### CCSS Standards

A logarithmic function is a constant multiple of a logarithm to some base, so it is a function given by $$f(x) = k \log_{a}(x)$$ where $$k$$ is any number and $$a$$ is a positive number (10, 2, or $$e$$ in this course). The graph of a typical logarithmic function is shown. Although the function grows very slowly, the graph does not have a horizontal asymptote.