http://www.ck12.org Chapter 3. Logs and Exponents
3.1 Exponential Functions
Here you will explore exponential functions as a way to model a special kind of growth or decay and you will learn
more about the numbere.
Exponential growth is one of the most powerful forces in nature. A famous legend states that the inventor of chess
was asked to state his own reward from the king. The man asked for a single grain of rice for the first square of the
chessboard, two grains of rice for the second square and four grains of rice for the third. He asked for the entire 64
squares to be filled in this way and that would be his reward. Did the man ask for too little, or too much?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/60994http://www.youtube.com/watch?v=7fpazNs1ZRE James Sousa: Graph Exponential Functions
Guidance
Exponential functions take the formf(x) =a·bxwhereaandbare constants. ais the starting amount whenx=0.
btells the story about the growth. If the growth is doubling thenbis 2. If the growth is halving (which would be
decay), thenbis^12. If the growth is increasing by 6% thenbis 1.06.
Exponential growth is everywhere. Money grows exponentially in banks. Populations of people, bacteria and
animals grow exponentially when their food and space aren’t limited.
Radioactive isotopes like Carbon 14 have something called a half-life that indicates how long it takes for half of the
molecules present to decay into other more stable molecules. It takes about 5,730 years for this process to occur
which is how scientists can date artifacts of ancient humans.
Example A
A mummified animal is found preserved on the slopes of an ice covered mountain. After testing, you see that exactly
one fourth of the carbon-14 has yet to decay. How long ago was this animal alive?
Solution: Since this problem does not give specific amounts of carbon, it can be inferred that the time will not
depend on the specific amounts. One technique that makes the problem easier to work with could be to create an
example scenario that fits the one fourth ratio. Suppose 60 units were present when the animal was alive at time
zero. This means that 15 units must be present today.
15 =a·( 1
2
)x60 =a·