Marketing: Undercounter Sales
Robert, a division manager for Giggles, a
major restaurant chain has heard rumors that
Selma, one of his store managers, is taking in
money without ringing up the sale.
Marketing: Undercounter Sales
After some investigative
Solution The bases, 4 and 8, can each be written as
a power of 2: 4 5 22 , 8 5 23 . 4a 5 8a11 Check 4a 5
8a11 22a 5 23a13 423 82311 2a 5 3a 1 3 423 822 2a
5 3 a 5 23 Answer a 5 23 EXAMPLE 3 Solve and
check: 3 1 7x21 5 10 Solution Add 23 to each side of
th
applications, the number e is a convenient choice to
use as the base. This is why the number e is also
called the natural base, and an exponential function
with e as the base is called a natural exponential
function. Recall that e 5 2.718281828 . . . Note
the following equation for x: 5 log2 x. Solution
Represent each side of the equation by y. Let y 5 .
Let y 5 log2 x. Then: Then: 3y 5 22 5 log2 x 3y 5 322
222 5 x y 5 22 5 x Answer x 5 1 4 1 4 1 9 log3 1 9
log3 1 9 "3 8 8 1 3 1 3 1 3 1 3 253 2 5 A251 2 B
and smaller but are always positive. We say that as x
approaches 2`, y approaches 0. The x-axis or the line
y 5 0 is a horizontal asymptote. EXAMPLE 1 a. Sketch
the graph of y 5 . b. From the graph, estimate the
value of , the value of y when x 5 1.2. c.
the graph from left to right, the values of y
decrease, that is, as x increases, y decreases. An
exponential function is a function of the form y 5 bx
where b is a positive number not equal to 1. If b . 1,
the function is an increasing function. If 0 , b
y 5 x. Ordered pairs of f21 (x) include , , (1, 0), (2, 1),
(4, 2), and (8, 3). The function y 5 logb x represents
logarithmic growth. Quantities represented by a
logarithmic function grow very slowly. For example,
suppose that the time it takes for a com
0.903089987. The logarithm is the exponent and
the antilogarithm is the power. In general: For logb x
5 y, x is the antilogarithm of y. EXAMPLE 1 Find log
23.75. Solution Use the key of the calculator. ENTER:
23.75 DISPLAY: Answer log 23.75 1.375663614
1.
logb c The log of a power is the exponent times the
log of the base. For example: 32 5 9 log3 9 5 2
(32 )3 5 3233 5 93 log3 (9)3 5 3 log3 9 5 3(2) 5 6
logb A c c B 1 9 log3 A 1 9 B 9 81 5 1 9 log3 A 9 81 B
9 81 c d Logarithmic Relationships 329
14411C08.p
all questions in this part. Each correct answer will
receive 2 credits. Clearly indicate the necessary
steps, including appropriate formula substitutions,
diagrams, graphs, charts, etc. For all questions in
this part, a correct numerical answer with no wo
be the value of the investment after n years if A0
dollars are invested at rate r per year. After 1 year:
A1 = A0 1 A0 r 5 A0 (1 1 r) A1 5 A0 (1 1 r) After 2
years: A2 5 A1 1 A1 r 5 A0 (1 1 r) 1 A0 (1 1 r)r 5 A0 (1
1 r)(1 1 r) 5 A0 (1 1 r)2 A2 5 A0 (1 1 r
negative if 0 , x , 1. Developing Skills In 314, find
the common logarithm of each number to the
nearest hundredth. 3. 3.75 4. 8.56 5. 47.88 6. 56.2
7. 562 8. 5,620 9. 0.342 10. 0.0759 11. 1 12. 10 13.
100 14. 0.1 In 1523, evaluate each logarithm to the
n
how many years will the investment have doubled
in value? Solution Let A dollars represent the value
of an investment of P dollars after t years at interest
rate r. Since interest is compounded yearly, A = P(1 1
r)t . If the value of the investment double
Therefore, the domain of y 5 bx is the set of real
numbers. When b . 1, as the negative values of x get
larger and larger in absolute value, the value of bx
gets smaller but is always positive. When 0 , b , 1, as
the positive values of x get larger and la
Writing About Mathematics 1. Show that the
formula A 5 A0 (1 1 r)n is equivalent to A 5 A0 (2)n
when r 5 100%. 2. Explain why, if an investment is
earning interest at a rate of 5% per year, the
investment is worth more if the interest is
compounded daily
rate r per interval of time, compounded n times per
interval, its value A after t intervals of time is: A 5 A0
If the increase or decrease is continuous for t units
of time, the formula becomes A 5 A0 ert A 1 1 r n B
nt A 5 A0(1 1 r)t b a x a b O x y O x
machine after 6 years? 46. Determine the common
solution of the system of equations: 15y 5 27x 5y 5
3x Exploration Prove that for all a: a. b.
CUMULATIVE REVIEW CHAPTERS 17 Part I Answer
all questions in this part. Each correct answer will
receive 2 credi
The half-life of an element is the length of time
required for the amount of a sample to decrease by
onehalf. If the weight of a sample is 1 gram and the
half life is t years, then: After one period of t
years, the amount present is 5 221 grams. After
two
animals can be said to increase continuously. If this
happens at a fixed rate per year, then the size of the
population in the future can be predicted. EXAMPLE
2 In a state park, the deer population was estimated
to be 2,000 and increasing continuously at
use it as a vacation fund. Each summer, she
withdraws an amount of money that reduces the
value of the fund by 7.5% from the previous
summer. How much will the fund be worth after the
tenth withdrawal? Solution Use the formula A 5 A0
(1 1 r)t with A0 5 10
graph of y 5 , locate the point (2a, b) on the graph of
the image. d. The equation of the image is y 5 or y
5 . Answers a. Graph b. 1.6 c. Graph d. y 5 We can
use the graphing calculator to find the values of the
function used to graph y 5 in Example 1. S
of the circle in standard form. c. Write the equation
of the line. d. Find the coordinates of the points at
which the line intersects the circle. 3 1 !24 1 2 !24
318 Exponential Functions 14411C07.pgs 8/12/08
1:51 PM Page 318 CHAPTER 8 319 CHAPTER TABLE
O
log3.5 log 10 4 log 5 2 log 17 5 30 340 Logarithmic
Functions Take the log of each side of the equation
and solve for the variable. 8x 5 32 log 8x 5 log 32 x
log 8 5 log 32 x 5 ENTER: 32 8 DISPLAY: 1.666666667
log(32)/log(8 Ans Frac 5/3 ENTER ENTER MATH
E
12 x 14411C08.pgs 3/3/09 2:13 PM Page 344 Check
Calculator check ln 12 2 ln x 5 ln 3 ln 12 2 ln 4 ln 3 ln
3 5 ln 3 Answer x 5 4 Note that in both methods
of solution, each side must be written as a single log
before taking the antilog of each side. EXAMPL