1. For the function 
,
a. Complete the table and use it to graph the function.
b. Describe the transformations that have been applied to
the graph of to 
obtain the graph .
c. What is the domain of ?
d. What is the range of ?
e. What are the coordinates of the y - intercept of
?
f. What is the equation of the asymptote for the graph of
?
2. a. Complete the table and graph the function
.
b. Using your knowledge of inverses, prepare a similar
table of values for the
inverse of the function .
c. Graph the inverse on the same grid as in problem 2(a) above.
Label each graph appropriately as f(x) or 
.
d. Answer the following questions for the inverse function:
What is the equation of the inverse function ?
What is the domain?
What is the range?
What are the coordinates of the x-intercept?
What is the equation of the asymptote?
Properties of Logarithms
Find the following natural logarithms with a calculator ; round to three decimal
places :
Answer the following questions by substituting the above
results:
1. Product Rule :
d. What do you conclude?
2. Quotient Rule:
d. What do you conclude?
3. Power Rule :
c. Make a general statement involving logarithms of
powers.
4. Composition of exp onential and logarithmic functions:
Exponential and logarithmic functions are inverses of each other. Inverse
functions
“undo” one another, so
. Use this fact for the
following
compositions.
Let 
and 
.
Find and simplify 
.
Find and simplify
.
Solving Exponential Functions
Solving exponential equations with bases which are powers of the same number:
A. Write both sides of the equation as powers of the same base.
B. Set the exponents equal to each other.
C. Solve for x.
Solving exponential equations with bases which are not
powers of the same number:
A. Isolate the exponential term, with a coefficient of one, on one side of the
equation.
B. Write the exponential equation as an equivalent logarithmic equation.
C. Solve for x.
Solving exponential equations with bases which are not
powers of the same number:
A. Isolate the exponential term(s), with a coefficient of one.
B. Take the “log” or “ln” of both sides.
C. Simplify using the Power Rule.
D. Solve for x.
Solving Logarithmic Functions
Solving logarithmic equations with log terms on only one side of the equation:
A. Make sure all log terms are on one side of the equation with non-log terms on
the
other side.
B. Use the properties of logarithms to write all sums and differences as a
single
logarithm.
C. Write the logarithmic equation as an equivalent exponential equation.
D. Solve for x.
E. Check the proposed solutions in the original equation.
Solving logarithmic equations with log terms with the same
base on both sides of the
equation:
A. Use the properties of logarithms to write each side as a single logarithm.
B. Set the arguments equal to each other.
C. Solve for x.
D. Check the proposed solutions in the original equation.
Applications of Exponential Functions
Exponential Growth
1. The population of Collin County, which follows the
exponential growth model,
increased from 264,036 in 1990 to 491,675 in 2000.
a. Find the exponential growth rate, k.
(Round answer to four decimal places.)
b. Write the exponential growth function.
c. What should the population be in 2012?
d. When should the population be 630,735?
e. How long will it take the population to double?
2. A midwestern city had a population of 950,000 in 2004. The population,
which follows the exponential model, decreased 7.63% per year.
a. Write the exponential function.
(Remember to change the rate to a negative decimal .)
b. What will the population be in 2012?
c. When will the population be 350,000?
Compound Interest
3. Lynsey invests $3000 in a bond trust that pays 8% interest compounded
semiannually. Her friend Lyla invests $3000 in a certificate of deposit that
pays 7.75% compounded monthly. Who has more money after 20 years,
Lynsey or Lyla?
OMIT 4. Given an initial investment of $5000 at 8% compounded continuously,
a. Find the amount after 3 years.
b. How many years would it take for the investment to grow to $7,000?
c. How long would it take for the investment to double?
Exponential Decay
5. The Lualailua Hills Quadrangle of the East Maui (Haleakala) volcano on the
island of
Maui in Hawaii is no longer active. To find out the date of the last eruption,
scientists
conducted a chemical analysis of samples from the volcano area. The samples
contained approximately 62.31% of its original carbon-14. When, to the nearest
year,
was the last eruption of the volcano? (Use 5715 years for the half-life of
carbon-14.)
6. A radioactive isotope, iron-52, used in the creation of medical images of
bone marrow,
has a half-life of 8.2 hours. If 8 microcuries are given to a patient, how many
microcuries
are left after 12 hours?
7. The function
can be used to find the number of milligrams D of a certain
drug that is in a patient’s bloodstream h hours after the drug has been
administered.
a. How many milligrams will be present after 2 hours?
b. When the number of milligrams reaches 1, the drug is to be administered
again.
After how many hours will the drug need to be administered?
OMIT Logistic Growth
8. On a college campus of 13,000 students, one student returned from spring
break with
a contagious virus. The spread of the virus is modeled by
where
y is the total number of students infected after t days.
a. How many students are infected after 7 days?
b. How many students are infected after 23 days?
c. How long will it be until 1500 students are infected?
