Download - Exponential and logrithmic functions
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EXPONENTIAL AND LOGARITHMIC FUNCTIONS
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EXPONENTIAL FUNCTIONIf 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
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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.
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The graph of the function y = bx
1
o
y
x
axisx:Asymptote Horizontal
none:erceptintx
1,0:erceptinty
,0:Range
,:Domain
xby
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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
x3y
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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
1y
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NATURAL EXPONENTIAL FUNCTION: f(x) = ex
1
o
y
x
axisx:Asymptote Horizontal
none:erceptintx
1,0:erceptinty
,0:Range
,:Domain
xexf
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LOGARITHMIC FUNCTIONFor 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 = byExamples of logarithmic functions: a) y = log3 x b) f(x) = log6 x
c) y = log2 x
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EXAMPLE 1: Express in exponential form:
204.0log )d
416log )c
532log )b
364log )a
5
2
1
2
4
749 )d
8127 )c
3216 )b
2166 )a
2
1
3
4
4
5
3
EXAMPLE 2: Express in logarithmic form:
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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.
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The graph of the function y = logb x
1o
y
x
axisy:Asymptote Vertical
none:erceptinty
1,0:erceptintx
,:Range
,0:Domain
xlogy b
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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
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EXAMPLE 2: Graph the function y = log1/3 x
1o
y
x
X 27 9 3 1 1/3 1/9 1/27
y -3 -2 -1 0 1 2 3
xlogy3
1
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PROPERTIES OF EXPONENTSIf a and b are positive real numbers, and m and n are rational numbers, then the following properties holds true:
mmm
mnnm
nmn
m
nmnm
baab
aa
aa
a
aaa
mnn mn
m
nn
1
mm
m
mm
aaa
aa
a
1a
b
a
b
a
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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:
x4xx2x
5x12x1x24x
273 d) 162
1 )c
84 b) 33 )a
5
210
24
32x )b
3x )a
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PROPERTIES OF LOGARITHMSIf M, N, and b (b ≠ 1) are positive real numbers, and r is any real number, then
xb
xblog
01log
1blog
NlogrNlog
NlogMlogN
Mlog
NlogMlogMNlog
xlog
xb
b
b
br
b
bbb
bbb
b
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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
52
6
3423
t
mnplog )c
py
xlog e) x3log )b
yx log d) xyz log )a
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EXAMPLE 2: Express as a single logarithm:
plog3
2nlog2mlog32log )c
nlog3mlog2 )b
3logxlog2xlog a)
5555
aa
222
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NATURAL LOGARITHMNatural logarithms are to the base e,
while common logarithms are to the base 10. The symbol ln x is used for natural logarithms.
2ln3xlnlne a) ln x EXAMPLE: Solve
for x:
1 elog e ln
x log x ln
e
e
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CHANGE-OF-BASE FORMULA
0.1 log c)
70 log b)
65 log a)
2
0.8
5
EXAMPLE: Use common logarithms and natural logarithms to find each logarithm:
bln
ln x x log or
b log
xlog x log b
a
ab
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Solving Exponential EquationsGuidelines: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
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Solving Logarithmic EquationsGuidelines: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:
2x264
9log )d
2
5xlog )b
4
x
25
4log )c 3
27
8log )a
8
34
5
2x
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EXAMPLE: Solve for x:
11xlog5xlog )f
xlog2xlog6xlog )e
25xlog25xlog d)
8ln x )c
3x25log b)
16 2x log 34 a)
77
222
52
5
2
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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.
isotope theof lifeHalfk
present isotope of amt. orig. reduce toit takes time t
isotope of amt. .origA
isotope of amt.present A : where2AA
o
k
t
o
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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.
t8.0e74991
000,15tf
lycontinuous compounded erestintPetA
yearper n times compounded erestintn
r1PtA
year 1for erestint simpler1PA
t r
t n