Chapter 4 – Exponential and Logarithmic Functions 4.1 - Exponential Functions.
STUDY GUIDE: Chapter 4: Exponential and Logarithmic Functions€¦ · STUDY GUIDE: Chapter 4:...
Transcript of STUDY GUIDE: Chapter 4: Exponential and Logarithmic Functions€¦ · STUDY GUIDE: Chapter 4:...
Name: _________________________________________ Date: ____________
STUDY GUIDE: Chapter 4: Exponential and Logarithmic Functions
Section 4.1
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 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.
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.
The exponential function describes the population of Mexico, f(x), in millions, x years after 1980.
\ Find Mexico's population in 2010.
4.
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 4.2Hints:
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.
5. Common logarithms are logarithms with base 10. In logarithmic notation, if the base is not given, it is
assumed to be 10.
Ex.:
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 394.
Additional Exercises: 1.
Write the exponential equation in its equivalent logarithmic form.
2. Evaluate the expression without using a calculator.
3.
Evaluate the expression without using a calculator.
4.
Graph both functions in the same coordinate plane.
Section 4.3Hints:
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.
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 4.4Hints:
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 4.3, 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.
Solve the exponential equation. Express your answer to two decimal places.
2.
Solve the exponential equation. Express your answer to two decimal places.
3.
.
Solve the logarithmic equation.
4.
Q: Solve the logarithmic equation. Express your answer to two decimal places.
Section 4.5Hints:
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.
Practice:
1. Suppose that you have $8000 to invest. Which investment will yield the greatest interest over 15 years?
A. 8.95% compounded quarterly
B. 8.85% compounded monthly
C. 9% compounded annually
D. 8.80% compounded continuously
2. Evaluate the expression.
A. 2/3 B. ½ C. –2 D. 2
3. Express the logarithmic form of the exponential equation.
A.
B.
C. D.
4. What is the x-intercept of the function?
A. 1
B. –2
C. 2
D. –1
5. Evaluate the expression.
A. 2.71 B. e C. 0.5 D. 1.36
6. Find the domain of the logarithmic function in interval notation.
A. B.
C.
D.
7. Write the expression as a single logarithm with a coefficient of 1.
A.
B.
C.
D.
8. Expand the expression and evaluate, if possible.
A.
B.
C.
D.
9. Solve for x.
A. 3.06
B. 2.01
C. 1.06
D. 2.92
10. Solve for x.
A. 13/2
B. {–2/3, 5}
C. {2/3}
D. {5}
11. Solve for x.
A. {5}
B. no solution
C. {–5, 1}
D. {5, –1}
12. Solve for x.
A. {0, ln 12}
B. {ln 6, ln 2}
C. {ln 7, ln 5}
D. {ln 4, ln 2}
13. How long will it take $1000 to double itself, if it is invested at a rate of 9% interest
compounded continuously?
A. 7.7 yr
B. 8.5 yr
C. 6.2 yr
D. 7.9 yr
14. Identify the graphs of f and g.
A.
B.
C.
D.
15. Write the expression as a single logarithm with a coefficient of 1.
A.
B.
C.
D.