Exponential and Logarithmic Functions
Why study exponential and logarithmic functions?
a. Exponential Functions
If we invest $1000 at 8% p.a., it grows to just under $5000 after 20 years.
There are many quantities that grow exponentially. Some examples are population, compound interest and charge in a capacitor.
An understanding of exponential growth is essential if you want to be comfortably rich later on...
The special thing about exponential growth is that the rate of growth increases as time increases. You can see this in the graph at right. The curve gets steeper and steeper as time goes on.
We can also have exponential decay (for example, radioactive decay).
Related Sections in
"Interactive Mathematics"
Exponents and Radicals, which is essential background before starting the current chapter.
Exponential form of a complex number
Differentiating the logarithmic function, Derivatives of exponential functions and Applications which shows how logarithms are used in calculus.
Integrating the exponential function, also part of calculus.
Differential equations: Electronics application with exponential decay and AIDS .
b. Logarithms
Logarithms were developed in the 17th century by the Scottish mathematician, John Napier. They were a clever method of reducing long multiplications into much simpler additions (and reducing divisions into subtractions). Young Johhny Napier had to help his dad, who was a tax collector. Johhny got sick of multiplying and dividing large numbers all day and devised logarithms to make his life easier.
The use of logarithms made trigonometry and many other fields of mathematics much simpler to calculate.
When calculus was developed later in the century, logarithms became central to many solutions. Today, logarithms are still important in many fields of science and engineering, even though we use calculators for most simple calculations.
You can see some applications in the "Related Sections" panel at right.
In this Chapter
- 1. Definitions: Exponential and Logarithmic Functions
- 2. Graphs of Exponential and Logarithmic Equations
- 3. Logarithm Laws
- 4. Logarithms to Base 10
- 5. Natural Logarithms (base e)
- Dow Jones Industrial Average (application)
- 6. Exponential and Logarithmic Equations
- World Population Live (application)
- 7. Graphs on Logarithmic and Semilogarithmic Paper
We begin the chapter with definitions of exponential and logarithmic functions »