Outcome C33: analyse and describe the characteristics of exponential and logarithmic functions
Originators: Hanna + Erika
Explanation: An exponential function is a function that changes exponentially according to the equation.
When graphed an exponential function forms a curve that either decreases (decays) or increases (growth) in the negative or postive direction exponentially.
Exponential functions generate geometric sequences which are found from dividing a y-term with the y-term before it to find the common ratio. Some things that are characterized by exponential change are nuclear reactions, economics, nuclear half-life, and carbon-14 dating. An exponential function has many forms. The most common form of the exponential function is the general form which is written as "y = a × b^((x-e)÷c) + d" The "a" being the starting y-value or vertical stretch, "b" being the common ratio, "c" being the unit of change within the set of x-values or horizontal stretch, "d" being the horizontal azymptote or vertical translation, "e" being the starting x-value or horizontal translation.
A logarithmic function is a function that is the inverse of an exponential function.
When graphed a logarithmic function is the reflection of the corresponding exponential graph over the line y=x.
Logarithmic functions can create its own personal scale called the logarithmic scale. This scale allows us to model real life situations such as Richter Scale which measures earthquakes and the Decibel level of sound which measures the intesity of sound. Like the exponential function a logarthmic function has its own general form. The general form of a logarthmic function can be written as "y=LOGbx" where the "LOG" is a base 10 log and can be evaluated with the change of base logarithmic equation, the "b" is the b-value of the logarithmic function's corresponding exponential function.
You can determine exponential or logarithmic functions in real life situations through identifying the horizontal translation, horizontal stretch, vertical translation, vertical stretch, and starting y and x-values of the function. After determining a function's characteristics it will be easy to graph from there on.
Sample Question: Julie works at a nuclear plant. During lunch one day, she accidentally drops 4g of caesium-137 on a group of oak trees out behind the plant. Caesium-137 is a radioactive isotope with a half-life of approximately 30 years.
a) Develop an equation that will determine the amount of Caesium-137 (y value) left behind after an x amount of years.
b) It is safe for people to be around Caesium-137 when the amount of Caesium-137 in the group of trees drops down to 0.5g. Using the equation previously made, determine how long it will take for it to be safe for workers to eat lunch by the tree once more.
Solution:
a) y = a × b^(x÷c) + d
a = 4 b = 2/4 = 0.5 c = 30
| Years (x) |
Caesium in grams (y) |
| 0 |
4 |
| 30 |
2 |
Therefore, the equation is y = 4(0.5)^(x/30)
b) y value is 0.5. goal is to find x value.
0.5 = 4(0.5)^(x/30)
step 1: 0.5/4 = 0.5^(x/30)
step 2: 0.125 = 0.5^(x/30)
log0.5^0.125 = x/30
step 3: log0.125/log0.5 = x/30
step 4: 3(30) = x
step 5: x = 90
Unfortunately, Julie will be dead before she can eat safely by the oak trees, as it will take 90 years.
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Comments (1)
Anonymous said
at 9:28 pm on Apr 29, 2007
I think I would fire Julie for safety violations! Very nice job on both the explaination and the sample problem.
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