Chapter 3: Exponential and Logarithmic Functions

Study Guide

Section 3.1
Hints:

1.   Functions whose equations contain a variable in the exponent are called exponential functions

Ex.:

2.  If the base of the exponential function has a base that is greater than one, the graph looks like the example below.

3.  If the base of the exponential function has a base that is between zero and one, the graph looks like the example below.

4.  The characteristics of exponential functions of the form are given on page 359.

5.  The irrational number, symbolized by the letter e, appears as the base in many application functions.  The value of e is approximately 2.71828. . .  The graph of the function, , is between the graphs of and , since e is between 2 and 3.

6.  Compound interest formulas are one common application of exponential functions.

is the formula for n compounding per year.

is the formula for continuous compounding.

 Additional Exercises: 1. Q:  Graph the exponential function. 2. Q:  Find the accumulated value of an investment of \$8,000 for 5 years at an interest rate of 6.4%, if the money is compounded quarterly.  Use the formula . 3. Q:  The exponential function describes the population of Mexico, f(x), in millions, x years after 1980.  Find Mexico's population in 2010. 4. Q:  Find the accumulated value of an investment of \$5,000 for 10 years at an interest rate of 3.2%, if the money is compounded continuously.  Use the formula .

Section 3.2
Hints:

1.  The inverse of the exponential function with a base b is a logarithmic function with base b.  The logarithmic function is defined as

2.  It is very important for you to be able to convert an exponential form, , to a logarithmic form, , and vice versa.

3.  Since logarithms are exponents, they have properties that can be verified using properties of exponents.  Some of the basic properties of logarithms are as follows.

1)

2)

3)

4)

4.  The figure below shows a graph of an exponential function and its inverse (a logarithmic function) when b>1.

The figure below shows a graph of an exponential function and its inverse (a logarithmic function) when 0<b<1.

Characteristics of the graphs of logarithmic functions are given on page 389.

5.  Common logarithms are logarithms with base 10.  In logarithmic notation, if the base is not given, it is assumed to be 10.

Ex.:

Common logarithms can be found using a calculator.  Properties of common logarithms are given on page 374.

6.  Natural logarithms are logarithms with base e.  The notation used for a natural logarithm is "ln".

Ex.:

Natural logarithms can be found using a calculator.  Properties of natural logarithms are given on page 376.

 Additional Exercises: 1. Q:  Write the exponential equation in its equivalent logarithmic form. 2. Q:  Evaluate the expression without using a calculator. 3. Q:  Evaluate the expression without using a calculator. 4. Q:  Graph both functions in the same coordinate plane.

Section 3.3
Hints:

1.   Properties of exponents correspond to properties of logarithms.  If you know the properties of exponents, this should help you remember the properties of logarithms.

2.  The product rule says that the logarithm of a product can be written as the sum of logarithms.

3.  The quotient rule says that the logarithm of a quotient can be written as the difference of logarithms.

When writing the sum of  logarithms the order of the terms does not matter; but, when writing the difference of  logarithms, watch the order of the terms.  You must subtract the logarithm of the denominator of the quotient from the logarithm of the numerator of the quotient.

4.  When you use the product, quotient, or power rule on a logarithmic expression, you are expanding the expression.  Think of the word "expand" as "to make larger".  When you use these properties, your final expression will be larger than your initial expression.  Sometimes it is necessary to use more that one property to completely expand the given logarithmic expression.

5.  When you write the sum or difference of two or more logarithmic expressions as a single logarithmic expression, you are condensing a logarithmic expression.  Think of the word "condense" as "to make smaller".  When you use these properties, your final expression will be smaller than your initial expression. The properties for condensing logarithms are the same as the properties for expanding.  The only difference is that sides of each property are reversed.  Compare the expanding properties on page 383 with the condensing properties on page 384

6.  A calculator will only find common logarithms and natural logarithms.  You can use the change-of-base property to find a logarithm with any other base.

 Additional Exercises: 1. Q:  Use properties of logarithms to expand the given logarithmic expression. 2. Q:  Use properties of logarithms to condense the given logarithmic expression. 3. Q:   Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places.

Section 3.4
Hints:

1.   An exponential equation is an equation containing a variable in an exponent.  You may use either natural or common logarithms and your calculator to solve exponential equations.  The steps for solving this type of equation using natural logarithms are

1)  Isolate the exponential expression.

2)  Take the natural logarithm on both sides of the equation.

3)  Simplify using one of the following properties:

or

4)  Solve for the variable.

2. A logarithmic equation is an equation containing a variable in a logarithmic expression.  If a logarithmic equation is in the form , you can solve it by rewriting the equation in the exponential form, .  In order to rewrite the logarithmic equation, you need a single logarithm whose coefficient is one.  Sometimes it is necessary to use the properties of logarithms, discussed in Section 3.3, to condense the logarithms in the equation.

3.  Logarithmic expressions are defined only for logarithms of positive real numbers.  You must always check your solutions in the original equation.  You should exclude any solution that produces the logarithm of a negative number or the logarithm of 0.

 Additional Exercises: 1. Q:  Solve the exponential equation.  Express your answer to two decimal places. 2. Q:  Solve the exponential equation.  Express your answer to two decimal places. 3. Q:  Solve the logarithmic equation. 4. Q:  Solve the logarithmic equation.  Express your answer to two decimal places.

Section 3.5
Hints:

1.  Predicting the behavior of variables can be done with exponential growth and decay models.  When a quantity grows or decays at a rate directly proportional to its size, this means that it grows or decays exponentially.

2.  The mathematical model for exponential growth or decay is given by

If k>0, the function models a quantity that is growing.  If k<0, the function models a quantity that is decaying.

3.  Many application problems require that you solve an exponential growth or decay model for the exponent in the problem.  This is the same process as solving an exponential function, which was discussed in the previous section.

4.  A logistic growth model is an exponential function used to model situations in which growth is limited.  The mathematical model for limited logistic growth is given by

,

where a, b, and c are constants, with c>0 and b>0.

5.  Sometimes it is helpful when working with an exponential model to express it in Base e.  This can be done by using the fact that is equivalent to .

 Additional Exercises: 1. Q:  The exponential growth model given describes the population of the United States, A, in millions, t years after 1970.  When will the U. S. population reach 375 million?  Round your answer to the nearest whole year. 2. Q:  The logistic growth function describes the population, f(t), of an endangered species of birds t years after they are introduced to a non-threatening habitat.  How many birds are expected in the habitat after 15 years?  Round your answer to the nearest whole number.

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