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Feb 20, 2024
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Uploaded by LieutenantSalmonPerson861
10/9/23, 6:52 AM
Topic 1.3c: MATHX402-032 Math for Management
https://onlinelearning.berkeley.edu/courses/2072192/pages/topic-1-dot-3c?module_item_id=96528771
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Topic 1.3c
Return to Module 1
(https://onlinelearning.berkeley.edu/courses/2072192/pages/module-01)
Topic 1.3c Using Exponential functions to Model
Growth and Decay
These 5 generations of Bunnies show a pattern of Exponential growth. How do you know? Click to
reveal.
The pattern 2, 4, 8, 16, 32 is a pattern of sequential growth of exponents on the base 2: , , ,
, .
Continuous growth and the use of e
Population growth and growth of money are common application problems where exponential and
logarithmic functions are used. Sometimes, this growth is discrete
(periodic, like, say, when interest
compounds once per year). Sometimes, the growth is continuous
(at a constant rate, like, say,
growth of bacteria, or bunnies). Continuous growth is modeled using , an irrational number, like ,
that has a value of about 2.718. Why? Briefly, it turns out that the limit, as gets very large, of the
expression:
...is , or 2.718. Go ahead, try it on your calculator, putting in increasingly large values for .
Remember our model of exponential growth of money, or compound interest, from Topic 1.3a is:
As we shorten the compounding periods (to, say, quarterly, monthly, daily, hourly…) and increase the
number of periods in the model (approaching continuous growth), the model is given as:
10/9/23, 6:52 AM
Topic 1.3c: MATHX402-032 Math for Management
https://onlinelearning.berkeley.edu/courses/2072192/pages/topic-1-dot-3c?module_item_id=96528771
2/4
Logically, you can see in this form, how the rate, , would get smaller as the compounding periods
shorten, and the number of periods, , would get very large, now resembling the basic formula
for the value of . At its limit (i.e., when approaches infinity), the value of the continuously growing
population, or continuously compounding principal, becomes:
I hope this has given you a theoretical basis for why is used in models of continuous growth, in
finance and otherwise. Don’t worry if you aren’t 100% clear on the calculations we’ve shown here.
You'll see applied examples here, in your textbook, in later modules (Module 8) and in your later
studies, that will make these equations, and the intuition behind them, more accessible to you.
Example 1.3c.1
How long will it take for $5,000 to grow to $10,000 at 8%, per year, compounded annually?
Because interest is paid annually, we use our interest compounding model for discrete growth:
Filling in the parts that we know:
Now, because we need to solve for a variable in the exponent, we know we need to use logs. We can
either:
(a) take the log of both sides:
And apply the Log of a Power property:
(b) or convert to log form:
And apply the change of base formula:
Check your answer by putting 9 back in the equation for :
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