Exponential functions.
is an exponential function with and
Notice how fast it grows. Similarly
Notice how fast it falls as x is increased!
Solving exponential equation. Easiest way is to seek expressions reduced to same bases.
Solve
Therefore, or
Exponential function to the natural base .
Example.
The value of 2.71828182845904523536 …
Sketch of the graph
Note that at , at
Therefore,
Logarithmic functions.
If then with and provided is +ve.
Notice,
The function looks like:
Some observations.
0.
1.
2. 2a.
3. work this out:
4.
5. (base change)
Two special logarithmic expressions:
a. With base 10 (
b. With base (
Therefore:
Some practice drill:
- Solving exponential equation.
Ex. 1 gives us
Therefore,
= 1.1609640
Ex. 2. Compound interest.
Principal invested = annual compound interest rate =
Investment period = years.
Total fund after years:
How long should one wait to double it?
Therefore,
Suppose, (5 percent) . Then years.
Suppose, compounding is done monthly instead. Then the equivalent monthly interest rate = 0.05/12=0.0041667. And, the answer changes to
months. < 14.21 years.
What about daily compounding? Interest 5% APR becomes
0.05/365 = 0.000136986. And, in days now becomes
days = 13.86 years.
Ex. 3 Bacteria growth in a colony. A colony can contain at most bacteria growing at the rate, say, per unit time. At time 0, a bacterium is introduced in the colony stacked up with 40 days old pizza crumbs. The bacteria population at time is given as
How long will it take for the bacteria colony to reach its two-third of its max population?
Let the time be hrs. Then the population equation becomes
How does this function look?
In terms of the bacterial model above, what might it correspond to?
What about ?
Ex 4. Endurance pill of the future for all couch potatoes. You don’t need to endanger your life by indulging in exercise.
Profile of ingesting a single pill.
Problem: How does one ensure a constant average level of medicine in blood stream?
Solution: Taking the medicine periodically. Let’s say the period is T. Then the profile should appear as
Suppose is ingested after every T units. Therefore,
=
…
At , the amount of medicine in the body, is
. is the desired level we want to maintain.