Exponential Functions
Exponential
note: migrated 18 Mar 2021
The word exponential makes this concept sound unnecessarily difficult.
A simple definition is that exponential models arise when the change in a quantity is proportional to the amount of the quantity. That is, the slope of an exponential function at any point is equal to the value of the function at any point multiplied by a number.
If a function is exponential, the relative difference between any two evenly spaced values is the same, anywhere on the graph. This is similar to linear functions where the absolute difference between any two values separated by the same x-axis distance is the same.
If you plot something on a log axis, you will notice that the distance between any two numbers with the same ratio is the equal. This preservation of the size of a ratio is the key feature of logarithms and exponentials.
Concepts
Euler's Number
Exponential notation
Place value
Slope proportional to the value
Doubling time
Euler's number
Exponential notation
Place Value
Recall from place value, that if we were using a base 2 system, each of these multiplications would be the same as shifting numbers one place to the left.
Slope proportional to function value
The exponential is defined as a function whose slope is proportional to its value.
Doubling Time
A property of an exponential function, is that in a given interval of time, the value increases by a fixed ratio.
Examples
Folding paper
How many times can you fold?
How can we express the number of pages by the number of folds?
Can we write out the pattern?
Rabbits
start with two
wait one year
now we have double (4)
wait another year
now we have eight (8)
how many in 5 years?
Money grows the same way
Start with $1000
Grow by 10%
Now $1100
Grow by 10%
Now $1210
Exponential growth
The rate of change is proportional to the total number
The doubling time is constant over the entire range
What things exhibit these characteristics?
Exponential growth
Populations
Disease spread
Credit card balances
Viruses
Definitions
Doubling
Doubling Time
Exponential growth means that if you measure how long it takes a quantity to double, the time to double from any current amount will be the same.
Constant Growth
This is in contrast to linear growth where if you wait for the amount to increase by a fixed amount, the amount will always increase by that fixed amount in that amount of time.
For linear growth, the slope of the function is a number that does not change along the function.
Which gets bigger faster?
Draw these out in your notebook to see
Exponential growth
Exponential Decay
What if instead of doubling every year, something fell by half each year?
Halving
Exponential Decay
Exponential decay
A quantity loses the same fraction of itself for a given time interval
Nuclear waste
Toxins in a body
The water in a stream (baseflow)
Logarithm
Inverse functions
Recall that the square root and the cube root were the inverse of the squared function and the cube function
Logarithm
The inverse of exponentiation is the logarithm
Properties of $e$
You have to specify your base on a computer (2, e, 10)
Logarithm
Inverse of exponential function
Logarithm
The logarithm in base 10 basically asks, if you are in base 10, how many digits?
Logarithmic scales
Musical pitch
Richter scale
Vision
Sound
In each of these, we perceive or use the logarithm.
This rule is a consequence of logarithms
Take log of both sides
Solving Problems
Analytical Approach
Use the logarithm as an inverse function.
Brute force approach
Miscellaneous Stuff
Unexpected connections
The number e and the natural logarithm are connected in various places to other mathematical numbers in fascinating ways.
The natural logarithm is the area under the 1/x curve
Definitions of e
Euler
Calculation examples
Most scripting languages and calculators use the
^
symbol forexponents.
2^2 = 4
.Python uses
**
for exponentiation.2**2 = 4
Some computer programs use
log
to mean the natural logarithm, notthe base 10 logarithm. Check carefully and see what your tool uses.
If you type
log(10)
and get 2.3 you are not using base 10.
Advanced Topics
I call these advanced topics mainly because the notation we use to express them is intimidating. The ideas behind them are simple and are unnecessarily obscured by our notation.
Differential equation
P is the population
r is the rate of growth
Any equation where the change is proportional to the population is exponential growth
This relates to our earlier definitions where the percent change is constant for a given time interval.
Differential equation
Linear growth has the changes is constant
Logistic function
Here the rate of change decreases as the population gets very large
This gives an s-shaped function.
You can see an example of this curve in the gangham style video statistics on youtube.
Time constant
This number expresses the time it takes for an exponential function to increase by a certain fraction.
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