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CHAPTER 2- Exponential &
Logarithmic Function
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Exponential Function
An exponential function has a “constant” which is raised to a variablepower, like y=2 x the basic shape of which when drawn, indicates a rapidly
increasing function.
Recall that a table of values consists of an ordered pairs of numbers (x,y)
representing a function or describing points in a coordinate plane. Consider
the rule y=2 x for the geometric progression 2,4,8,16,….
domin: x/x is R number
range: x/x is all positive R numbers
examples:
y=2 x
x -3 -2 -1 0 1 2 3 4
y 1/8 1/4 1/2 1 2 4 8 16
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b0=1
20=1
b-1= b ≠ 0
bm/n= bm
Any function raise to 0 is one
1 bm
n
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Theorem 2.1
if b > 0, b ≠ 1 & if y > 0, there existence one & only onereal number x such that
y= bx
Theorem 2.2
if b > 0, b ≠ 1 & if x, & x2 are two numbers such that bx1
=bx2 when x1 =x2
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25x=6 5x =? 36x+1=5 62x+2 =?
52x=6 54x= 36 62(x+1)=5 62x+2= 5
52(2x) =6 62x+2 =5
54x =36
63(x+2)= 53 2162x+2 = ?
66x+6= 125 63(x+2)=?
63(x+2)= 125
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Seatwork
Tell which of the following define an
exponential
1) y=-3x
2) y=3x
3) y=x2
+14) y=(5) -x
5) y=3x2
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Homework
Graph the following exponential function
1) y=3x
2) y=4x
3) y=5x
4) y=6x 5) y=7x
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Logarithmic Function
The logarithm of a number is the exponent by
which another fixed value, the base, must be raised to
produce that number. For example, the logarithm of
1000 to base 10 is 3, because 1000 is 10 to the power
3: 1000 = 10 × 10 × 10 = 103. More generally, if x = by ,then y is the logarithm of x to base b, and is written y =
logb( x ), so log10(1000) = 3.
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y=logb x x=by
y=log3 x x=3y
x 1/3 1 3
y -1 0 1
y
x
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1) 0 function x=1 logb 1=0
2) 1 function x=b log3 3=1
3) X > 1 positive
4) X < 1 negative
5) Increasing fx
1) loga (AB)= loga A+B+loga
2) loga (AB)= loga A+B+loga
3) loga (AB)= loga A+B+loga
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Log3 (9) (27) = log3 9+ log3 27=5 y=log5 1/125 3x =1/125
3x = 9 3x = 27
3x = 32 3x = 33
x= 2 x= 3
Log3 63-log3 7=2 log10 (3 √ 125) (0.0356) 2 3
Log3 3(63/7) (36.3)(√5.42)
Log3 9 3 log10 3√ 125+ 2log10 .0356
3x
=9 log10 36.2- log10 √ 5.423x = 32 1/3 log10 125+ 2log10 0.0356
x= 2 1/ 3 log10 36.2-1/2log10 5.42
log10 125+ 6log 0.0356
3 log10 36.2-3/2log10 5.42
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Seatwork
Convert log to exponential
1) log4 16=2
2) log3 18=4
3) log2 1/32=-54) logb3=2
5) log1/2 2=8
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Common Logarithmic
The common logarithm is the logarithm with base 10. It’scalled the common logarithm because we use abase-10
number system in our daily lives. This type of logarithm
is often simply written as log(x). log(x) gives you the
number y such that 10y = x.
The common logarithm has all the same usefulproperties
as any other logarithm:
1) log(xy) = log(x) + log(y)
2) log(x/y) = log(x) – log(y)
3) log(xy) = y * log(x)
4) log 0.00635= -2.2041
5) log 6.25= 0.7959
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Finding the common logarithm log M
Log M= log N+log10k
= log N+k
= k+log N=k+M
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Seatwork
1)What is log(1000)?
2)What is log(0.1)?
3) What is log(1)?4) What is log(25) + log(4)?
5) What is log(50) – log(5)?
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Homework
1)Compute 4√ 3.528 using log
2)Compute 3√ 326 (0.00253)
(2.03)2
3)Compute 2√ 4.15 3using log
4)Compute 5√ 2.122using log
5)Compute 3√ 2.105 using log
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Natural Logarithmic
The natural logarithm is the logarithm to the base e,
where e is an irrational and transcendental constant
approximately equal to 2.718281828. The natural
logarithm is generally written as ln x , loge x or
sometimes, if the base of e is implicit, as simply log x .
Parentheses are sometimes added for clarity, giving
ln( x ), loge( x ) or log( x ). This is done in particular when the
argument to the logarithm is not a single symbol, in order
to prevent ambiguity
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e y =M then y= ln M
e y =21 y=21
y= 3.0445Theorem 2.3
if a & b are bases & m > 0
Logb M= logaMloga b
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Find the value of loge 10
Log10 = 1 = 2.203
Loge 0.434Find the value of loge 8
Log10 = 0.9031 = 1.893
Loge 0.4771
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Seatwork
Find the value of each of the following.
1) log4 48
2) loge 573) ln 1/3.6
4) ln 1/0.089
5) log2 6
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Homework
Find approximately the values of y
1) y= log8 42
2) y= loge 3523) log7 y=3
4) loge y=2.42
5) ln y=1.50
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Solution of Exponential &
Logarithmic EquationIn previous topic we learned about the exponential
and logarithmic functions, studied some of their
properties, and learned some of their applications. In this
topic we show how to solve some simple
equations which contain the unknown either as an
exponent (exponential equation) or as the argument
of a logarithmic function.
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Example: 4x+1 = 3
2x = 7 log4 4x+1 = log4 3
log2 2x = log2 7 (x + 1) log4 4 = log4 3
x log2 2 = log2 7 x + 1 = log4 3
x = log2 7 2.807 x = log4 3 1
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3x-3 = 5 e2x-3ex + 2 = 0
x - 3 = log3 5 (e x)2x -3e x + 2 = 0
x = log3 5 + 3 (ex
-1)(ex
- 2) = 0e x - 1 = 0
e x = 1
x = 0
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Solve the equation 7x+1 =32x-3 for x
7x+1 =32x-3
(x+1) log 7= (2x-3) log 3
Xlog 7+ 3log 3= 2xlog 3 -xlog7
Log 7 +3log 3= x(2 log 3-log 7)2log 3-log 7 2log 3-log 7
X= log 7 + 3 log 3
2log 3- log 7
= 0.8451+ 1.4310.9542-0.8451
= 2.2764
.1091
=20.8653
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Seatwork
1) Solve the equation 3x log 7+ xlog 5 = log 35 + log 5 for x
2) Solve the equation 72x-7x-1 = 0 for x
3) Solve the equation 3xex + x2ex = 0
4) Solve the equation log2(x + 2) = 55) Solve the equation log7(25 -x) = 3
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Homework
1) Solve the equation ex +3-x =2 for x
2) Solve the equation 2x-1 +2-x =4 for x
3) Solve the equation log(3x+2)= log(x-4)+1 for x
4) Solve the equation log(2x+4)= log(x-1/2)+2 for x
5) Solve the equation log(4x-1 -1)= log (2x-2 -1)-2 for
x
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