This document discusses exponential and logarithmic functions. It defines exponential functions as functions of the form f(x) = bx where b is the base, and defines logarithmic functions as the inverses of exponential functions. Properties of exponential and logarithmic functions are presented, including their domains, ranges, and asymptotes. Examples of graphing common exponential and logarithmic functions are shown. Methods for solving exponential and logarithmic equations are also provided.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Had to make this dumb powerpoint for my algebra II class and I put a lot of work into it for some reason... so yeah it's just been sitting on my laptop doing nothing and I thought why not upload this to help other people? So yeah, hope you guys find it useful...
During this webinar, we will discuss the following:
• General Facebook demographics (Age, Location, Education)
• The Facebook Connect Platform
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logarithmic, exponential, trigonometric functions and their graphs.pptYohannesAndualem1
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1) Use properties of logarithms to expand the following logarithm.docxdorishigh
1) Use properties of logarithms to expand the following logarithmic expression as much as possible.
Logb (√xy3 / z3)
A. 1/2 logb x - 6 logb y + 3 logb z
B. 1/2 logb x - 9 logb y - 3 logb z
C. 1/2 logb x + 3 logb y + 6 logb z
D. 1/2 logb x + 3 logb y - 3 logb z
2) Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.
2 log x = log 25
A. {12}
B. {5}
C. {-3}
D. {25}
3) Write the following equation in its equivalent logarithmic form.
2-4 = 1/16
A. Log4 1/16 = 64
B. Log2 1/24 = -4
C. Log2 1/16 = -4
D. Log4 1/16 = 54
4) Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log2 96 – log2 3
A. 5
B. 7
C. 12
D. 4
5) Use the exponential growth model, A = A0ekt, to show that the time it takes a population to double (to grow from A0 to 2A0 ) is given by t = ln 2/k.
A. A0 = A0ekt; ln = ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
B. 2A0 = A0e; 2= ekt; ln = ln ekt; ln 2 = kt; ln 2/k = t
C. 2A0 = A0ekt; 2= ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
D. 2A0 = A0ekt; 2 = ekt; ln 1 = ln ekt; ln 2 = kt; ln 2/k = toe
6) Find the domain of following logarithmic function.
f(x) = log (2 - x)
A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)
7) An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?
A. Approximately 7 grams
B. Approximately 8 grams
C. Approximately 23 grams
D. Approximately 4 grams
8) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2 y) / z2
A. 2 logb x + logb y - 2 logb z
B. 4 logb x - logb y - 2 logb z
C. 2 logb x + 2 logb y + 2 logb z
D. logb x - logb y + 2 logb z
9) The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.
A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)
10) Approximate the following using a calculator; round your answer to three decimal places.
3√5
A. .765
B. 14297
C. 11.494
D. 11.665
11) Write the following equation in its equivalent exponential form.
4 = log2 16
A. 2 log4 = 16
B. 22 = 4
C. 44 = 256
D. 24 = 16
12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
31-x = 1/27
A. {2}
B. {-7}
C. {4}
D. {3}
13) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2y)
A. 2 logy x + logx y
B. 2 logb x + logb y
C. logx - logb y
D. logb x – ...
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The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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2. EXPONENTIAL FUNCTION
If x and b are real numbers such that b > 0
and b ≠ 1, then f(x) = bx is an exponential
function with base b.
Examples of exponential functions:
a) y = 3x b) f(x) = 6x c) y = 2x
Example: Evaluate the function y = 4x at the given
values of x.
a) x = 2 b) x = -3 c) x = 0
3. PROPERTIES OF EXPONENTIAL FUNCTION y =
bx
• The domain is the set of all real numbers.
• The range is the set of positive real numbers.
• The y – intercept of the graph is 1.
• The x – axis is an asymptote of the graph.
• The function is one – to – one.
4. The graph of the function y = bx
1
o
y
x
axisx:AsymptoteHorizontal
none:erceptintx
1,0:erceptinty
,0:Range
,:Domain
x
by
5. EXAMPLE 1: Graph the function y = 3x
1
X -3 -2 -1 0 1 2 3
y 1/27 1/9 1/3 1 3 9 27
o
y
x
x
3y
6. EXAMPLE 2: Graph the function y =
(1/3)x
1
X -3 -2 -1 0 1 2 3
y 27 9 3 1 1/3 1/9 1/27
o
y
x
x
3
1
y
7. NATURAL EXPONENTIAL FUNCTION: f(x) = ex
1
o
y
x
axisx:AsymptoteHorizontal
none:erceptintx
1,0:erceptinty
,0:Range
,:Domain
x
exf
8. LOGARITHMIC FUNCTION
For all positive real numbers x and b, b ≠ 1,
the inverse of the exponential function y = bx is
the logarithmic function y = logb x.
In symbol, y = logb x if and only if x = by
Examples of logarithmic functions:
a) y = log3 x b) f(x) = log6 x c) y = log2 x
10. PROPERTIES OF LOGARITHMIC FUNCTIONS
• The domain is the set of positive real
numbers.
• The range is the set of all real numbers.
• The x – intercept of the graph is 1.
• The y – axis is an asymptote of the graph.
• The function is one – to – one.
11. The graph of the function y = logb x
1o
y
x
axisy:AsymptoteVertical
none:erceptinty
1,0:erceptintx
,:Range
,0:Domain
xlogy b
12. EXAMPLE 1: Graph the function y = log3 x
1
X 1/27 1/9 1/3 1 3 9 27
y -3 -2 -1 0 1 2 3
o
y
x
xlogy 3
13. EXAMPLE 2: Graph the function y = log1/3 x
1
o
y
x
X 27 9 3 1 1/3 1/9 1/27
y -3 -2 -1 0 1 2 3
xlogy
3
1
14. PROPERTIES OF EXPONENTS
If a and b are positive real numbers, and m and n are rational numbers, then
the following properties holds true:
mmm
mnnm
nm
n
m
nmnm
baab
aa
a
a
a
aaa
m
nn mn
m
nn
1
m
m
m
mm
aaa
aa
a
1
a
b
a
b
a
15. To solve exponential equations, the following property can be used:
bm = bn if and only if m = n and b > 0, b ≠ 1
EXAMPLE 1: Simplify the
following:
EXAMPLE 2: Solve for
x:
x4xx
2x
5x12x1x24x
273d)16
2
1
)c
84b)33)a
5
2
10
24
32x)b
3x)a
16. PROPERTIES OF LOGARITHMS
If M, N, and b (b ≠ 1) are positive real numbers,
and r is any real number, then
xb
xblog
01log
1blog
NlogrNlog
NlogMlog
N
M
log
NlogMlogMNlog
xlog
x
b
b
b
b
r
b
bbb
bbb
b
17. Since logarithmic function is continuous and
one-to-one, every positive real number has a
unique logarithm to the base b. Therefore,
logbN = logbM if and only if N = M
EXAMPLE 1: Express the ff. in expanded form:
24
35
2
5
2
6
34
23
t
mnp
log)c
py
x
loge)x3log)b
yxlogd)xyzlog)a
18. EXAMPLE 2: Express as a single logarithm:
plog
3
2
nlog2mlog32log)c
nlog3mlog2)b
3logxlog2xloga)
5555
aa
222
19. NATURAL LOGARITHM
Natural logarithms are to the base e, while
common logarithms are to the base 10. The
symbol ln x is used for natural logarithms.
2ln3xlnlnea) ln x
EXAMPLE: Solve for
x:
1elogeln
xlogxln
e
e
21. Solving Exponential Equations
Guidelines:
1. Isolate the exponential expression on one side of
the equation.
2. Take the logarithm of each side, then use the law
of logarithm to bring down the exponent.
3. Solve for the variable.
EXAMPLE: Solve for
x:
06ee)d
4e)c
20e8)b
73)a
xx2
x23
x2
2x
22. Solving Logarithmic Equations
Guidelines:
1. Isolate the logarithmic term on one side of the
equation; you may first need to combine the
logarithmic terms.
2. Write the equation in exponential form.
3. Solve for the variable.
EXAMPLE 1: Solve the
following:
2x2
64
9
log)d
2
5
xlog)b
4
x
25
4
log)c3
27
8
log)a
8
34
5
2x
24. Application: (Exponential and Logarithmic Equations)
•The growth rate for a particular bacterial culture can be
calculated using the formula B = 900(2)t/50, where B is the
number of bacteria and t is the elapsed time in hours. How
many bacteria will be present after 5 hours?
• How many hours will it take for there to be 18,000
bacteria present in the culture in example (1)?
• A fossil that originally contained 100 mg of carbon-14 now
contains 75 mg of the isotope. Determine the approximate
age of the fossil, to the nearest 100 years, if the half-life of
carbon-14 is 5,570 years.
isotopetheoflifeHalfk
presentisotopeofamt.orig.reducetoit takestimet
isotopeofamt..origA
isotopeofamt.presentA:where2AA
o
k
t
o
25. 4. In a town of 15,000 people, the spread of a rumor that the
local transit company would go on strike was such that t
hours after the rumor started, f(t) persons heard the
rumor, where experience over time has shown that
a) How many people started the rumor?
b) How many people heard the rumor after 5 hours?
5. A sum of $5,000 is invested at an interest rate of 5% per
year. Find the time required for the money to double if the
interest is compounded (a) semi-annually (b) continuously.
t8.0
e74991
000,15
tf
lycontinuouscompoundederestintPetA
yearpern timescompoundederestint
n
r
1PtA
year1forerestintsimpler1PA
tr
tn