Download MTH108 lec 18

MTH108 Business Math 1
Lecture 18
Chapter 7
Exponential and
Logarithmic Functions
Review
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Properties of exponents and radicals
Exponential functions
Classes of exponential functions
Graphs of exponential functions
Conversion to natural base e
Logarithmic functions
Properties of logarithms
Review
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Solving logarithmic equations
Graphs of logarithmic functions
Characteristics of logarithmic functions
Some applications
Today’s Topic
• Some applications of exponential functions
• Some applications of logarithmic functions
• Solving exponential and logarithmic equations
Applications of exponential functions
Exponential functions have particular application to
growth process and decay process.
Growth process: population growth, appreciation in the
value of assets, … exponential growth process
Decay process: declining value of certain assets such as
machinery, decline in the efficiency of machine, …
exponential decay process.
Both process are usually stated in terms of time.
Compound Interest: the interest earned by an invested
amount of money is reinvested so that it too earn
interest, i.e. the interest is converted into principal
and hence there is “interest on interest”.
e.g. 100 dollar is invested at the rate of 5 percent
compounded annually.
At the end of first year
Compound Interest:
More generally, if a principal amount of P dollars is
invested at the rate of 100r percent compounded
annually, then compound amount will be:
After 1 year:
Compound Interest:
Compound Interest (Continuous compounding):
When compounding of interest is more than once a
year, the previous equation can be restated as:
Compound Interest (Continuous compounding):
When compounding of interest is continuous means
compounding is occurring all the time, i.e. there are
infinite number of compounding periods each year.
Then,
Population model: Consider the general function of the
exponential growth process
Population model (contd.):
How long will it take for the value of the function to
increase by some multiple, e.g. how long will it take
for the population to double?
Decay functions (price of a machinery):
Consider the general form of the exponential decay
function as:
Logarithmic Functions
Consider the case of population growth. Recall that the
exponential function for this case is:
Population growth case:
Decay case: Recall the general form of the exponential
decay process as:
Suppose the amount of radioactive substance is
reduced to half when k= 4 %
Solving logarithmic and exponential equations
Recall that a logarithmic equation is an equation that
involves the logarithm of an expression containing an
unknown. e.g.
An exponential equation has the unknown appearing
in an exponent, e.g.
In many functions, if
However, logarithmic and exponential functions have
these properties, i.e.
We have used these facts and now we will see them in
detail.
1)
Oxygen composition An experiment was conducted
with a particular type of small animal. The logarithm
of the amount of the oxygen consumed per hour was
determined for a number of the animals and was
plotted against the logarithms of the weights of the
animals. It was found that
Oxygen composition
Demand equation
The demand equation for a product is
Express q in terms of p.
Demand equation
Solve
Predator-Prey relation
Consider an equation of the form
Soln.:
Predator-Prey relation
Solve
Verify
Summary
•
•
•
•
•
•
•
•
•
Properties of exponents and radicals
Exponential functions
Classes of exponential functions
Graphs of exponential functions
Conversion to natural base e
Logarithmic functions
Properties of logarithms
Solving logarithmic equations
Graphs of logarithmic functions
Summary
•
•
•
•
•
Characteristics of logarithmic functions
Some applications
Some applications of exponential functions
Some applications of logarithmic functions
Solving exponential and logarithmic equations