UNIT OVERVIEW
This unit builds on your work with inverse functions. You learned that the exponential function has an inverse. That inverse is called the logarithmic function.
A logarithm answers one simple question: "What power do I need?"
When you ask how many years it takes for your savings to double, or how long before a population grows tenfold, you are asking a logarithmic question.
By the end of this unit, you will be able to:
- Convert between exponential and logarithmic forms
- Understand why logarithms work the way they do
- Sketch and interpret graphs of for different base values
- Apply the four logarithmic laws to simplify expressions and solve equations
- Solve problems involving compound interest, population growth, and radioactive decay
Real-Life Applications
Finance: Logarithms tell you how long it takes for an investment to reach a target amount.
Sound: The decibel scale is logarithmic. A sound at 120 dB is one million times more intense than one at 60 dB.
Earthquakes: The Richter scale is logarithmic. A magnitude 7 earthquake releases about 31.6 times more energy than a magnitude 6.
Chemistry: The pH scale measures acidity logarithmically. Each unit change represents a tenfold change in hydrogen ion concentration.
Archaeology: Carbon dating uses logarithms to determine the age of fossils.