MTH108 Business Math 1

Lecture 18

Chapter 7

Exponential and Logarithmic Functions

Review

• 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

• 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 equationThe demand equation for a product is Express q in terms of p.

Demand equation

Solve

Predator-Prey relationConsider 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