Unit 8: Exponential and Logarithmic Equations
Now that you know all about exponential and logarithmic functions, you can learn how to solve equations that involve them. You will use the properties of logarithms and exponentials and the fact that they are inverses to solve equations. You will also explore some applied problems involving exponential and logarithmic equations. Finally, you will learn to fit exponential and logarithmic models to data.
Completing this unit should take you approximately 3 hours.
Upon successful completion of this unit, you will be able to:
- apply properties of logarithms to rewrite and simplify logarithmic expressions, including the product, quotient, and power rules and the change-of-base formula;
- solve exponential equations using like bases and logarithms;
- solve logarithmic equations using the definition of a logarithm and the one-to-one property of logarithms;
- solve applications of exponential and logarithmic equations, including half-life and Newton's law of cooling;
- differentiate between exponential growth and decay;
- identify the key components of a logistic growth model; and
- build exponential and logarithmic regression models from data, and assess their fit.
8.1: Solving Exponential Equations
Getting the variable out of the exponent can be tricky, but you will learn the basics in this section.
The next step in solving exponential equations involves using their inverse, the logarithm. You will work with several different bases and even with exponents that are expressions.
8.2: Solving Logarithmic Equations
In this unit, you will explore the techniques for solving logarithmic equations. We will begin by using the definition of a logarithm to "undo" it. Then, we will work up to more complex techniques.
Now, we will solve applied problems that involve half-life and the radioactive decay of chemical elements.
8.3: Exponential and Logarithmic Models
Did you know that you can predict the temperature of a cup of hot tea after it sits for 20 minutes? This section will take a deeper look at applications of exponential functions and the behaviors they model in the real world around us. We will explore half-life, radioactive decay, and Newton's law of cooling.
The spread of a virus such as COVID-19 depends on how many people have the virus and how many people are left in the population to which the virus can spread. This behavior can be modeled using a logistic growth model. In this section, you will learn about the different components of a logistic growth model and what behaviors it is used to model.
8.4: Fitting Exponential and Logarithmic Models to Data
Now, we will practice building models using datasets. Here, you will see a short video on using an online graphing calculator to do the calculations required. Billions of data points are collected every year in fields from consumer behavior to weather. Fitting this data to a model allows us to explore the behaviors we observe around us meaningfully. The first model you will build is logarithmic.