LOGARITHMIC FUNCTIONS Presented by: AMEENA AMEEN MARYAM BAQIR FATIMA EL MANNAI KHOLOOD REEM IBRAHIM...
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Transcript of LOGARITHMIC FUNCTIONS Presented by: AMEENA AMEEN MARYAM BAQIR FATIMA EL MANNAI KHOLOOD REEM IBRAHIM...
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LOGARITHMIC FUNCTIONSPresented by:
AMEENA AMEEN MARYAM BAQIRFATIMA EL MANNAIKHOLOODREEM IBRAHIMMARIAM OSAMA
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Content:• Definition of logarithm • How to write a Logarithmic form
as an Exponantional form• Properties of logarithm• Laws of logarithm• Changing the base of log• Common logarithm
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• Binary logarithm• Logarithmic Equation• The natural logarithm• Proof that d/dx ln(x) =1/x• Graphing logarithmic functions.
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AMEENA
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Definition of Logarithmic Function
The power to which a base must be raised to yield a given number
e.g.the logarithm to the base 3 of 9, or
log3 9, is 2, because 32 = 9
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The general form of logarithm:
• The exponential equation could be written in terms of a logarithmic equation as this form
• a^ y = Х Loga x = y
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Example of logarithms:
416log2
1
5243log3
216
1log4
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Common logarithms:
The two most common logarithms are called (common logarithms)
and( natural logarithms).Common logarithms have a base of 10
log x = log10x , and natural logarithms have a base
of e. ln x =logex
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Exponential form:-
3^3=27
2^-5=1/32
4^0=1
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MARYAM.B
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Properties of Logarithm
• because
• because
• becausexa xa log
1log aa
01log a 10 a
aa 1
xaxa
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Property1: loga1=0 because a0=1
• Examples:
• (a) 90=1
• (b) log91=0
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Property 2: logaa=1 because a1=a
• Examples:
• (a) 21=2
• (b)log22=1
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Property 3:logaax=x because ax=ax
• Examples:
• (a) 24=24
• (b) log224=4
• (c) 32=9 log39=2log332=2
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Property4:blogbx=x
• Example:
• 3log35=5
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FATIMA
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There are three laws of logarithms:
nmmn aaa logloglog
nmn
maaa logloglog
mnm an
a loglog
1
2
3
Logarithm of products
Logarithm of quotient
Logarithm of a power
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Remember these laws:
01log a
1log aa
1
2
The log of 1 is always equal to 0 but the log of
a number which is similar
to the base of log is always
equal 1
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nmmn aaa logloglog
Example:
3log5log aa
)35(log a
15loga
Transform the addition into multiplication
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nmn
maaa logloglog
Example2:7log35log aa
)7
35(loga
5loga
Transforming the subtraction into division
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mnm an
a loglog
Example3:
8log24log3 aa
4096log
64log2
64log64log
a
a
aa
4log3 a
3)4(loga
8log2 a
2)8(loga
The form of
Will be changed into
And the same for
Will be
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Solve it yourself!
3log24log5log2 aaa
22 3log4log5log aaa
9
111log
9
425log
9
425log
a
a
a
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KHOLOOD
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Let a, b, and x be positive real numbers such that and (remember x must be greater than 0). Then can be converted to the base b by the formula
Changing the base:
xba log
ba x
ba cx
c loglog
bax cc loglog
a
bx
c
c
log
log
let
Divide each side by
Take the base-c logarithm of each side
Power rule
aclog
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a
bb
c
ca log
loglog
If a and b are positive numbers not equal to 1 and M is positive, then*
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If the new base is 10 or e, then:*
)ln(
)ln(
)log(
)log()(log
a
b
a
bba
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Common logarithm:
In mathematics,the common logarithm is the logarithm with base 10. It is also
known as the decadic logarithm, [] .
Examples:
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Binary logarithm:
The binary logarithm is the logarithm for base 2. It is the inverse function of .
Examples:
n2
n2log
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Binary logarithm:
In mathematics, the binary logarithm is the logarithm for base 2. It is the inverse function of .
Examples:
n2n2log
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REEM
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The Nature of Logarithm
Is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.
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The Nature of LogarithmThe natural logarithm
can be defined for all positive real numbers x as the area under the curve y = 1/t from 1 to x, and can also be defined for non-zero complex numbers.
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The Nature of LogarithmThe natural logarithm function can
also be defined as the inverse function of the exponential function, leading to the identities:
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Logarithm Equation
Logarithmic equations contain logarithmic expressions and constants. When one side
of the equation contains a single logarithm and the other side contains a constant, the
equation can be solved by rewriting the equation as an equivalent exponential
equation using the definition of logarithm.
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For Example
Property of Logarithms:
Definition of Logarithm
8 = x- 7x Simplify
0 = x- 7x – 8 Write quadratic equation in standard form
0 = (x – 8)(x + 1) Solve by factoring
x – 8 = 0 or x + 1 = 0x = 8 or x = -1
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Substitute the solution
8 for x
Substitute the solution –1 for x
Subtract Subtract
3 + 0 = 3 Because ;
because
The number -1 does not check, since negative numbers do not have logarithm
s 3=3
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Proof that d/dx ln(x) = 1/xThe natural log of x does not equal 1/x, however the derivative
of ln(x) does:
The derivative of log(x) is given as:d/dx] log-a(x) [ = 1 / (x * ln(a))where "log-a" is the logarithm of base a.
However, when a = e (natural exponent), then log-a(x) becomes ln(x) and ln(e) = 1:
d/dx] log-e(x) [ = 1 / (x * ln(e))
d/dx] ln(x) [ = 1 / (x * ln(e))
d/dx] ln(x) [ = 1 / (x * 1)
d/dx] ln(x) [ = 1 / x
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MARIAM QAROOT
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Graphing logarithms is a piece of cake!!
• Basics of graphing logarithm
• Comparing between logarithm and exponential graphs
• Special cases of graphing logarithm
• The logarithm families.
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Graphing Basics:
• The important key about graphing in general, is to stick in your mind the bases for this graph.
• For logarithm the origin of its graph is square-root graph..
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01log by(b,1)
b1
1
Before graphing y= logb (x) we can start first with knowing the following:
The logarithm of 1 is zero (x=1), so the x-intercept is always 1, no matter what base of log was.For example if we have: b = 2 power 0 = 1 b = 3 power 0 = 1 b = 4 power 0 = 1
Values of x between 0 and 1 represent the graph below the x-axis when:
10 xFractions are the values of the negative powers.
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Examples on graphing logarithm:• EXAMPLE ONE
Graph y = log2(x).First change log to exponent
form:X=2 power y, then start with
a T-chart
X Y= LOG 2(X)
0.125 log2(0.125) = –3
0.5 log2(0.5) = –1
1 log2(1) = 0
2 log2(2) = 1
4 log2(4) = 2
8 log2(8) = 3
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• EXAMPLE two:
GraphFirst change ln into logarithm
form:Loge (x)Then change to exponential
form:X= e power y..Now draw you T-
chart
xy ln
X Y= loge (x)
0.13 -2
0.36 -1
1 0
2.71 1
7.38 2
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• EXAMPLE two:Graph y = log2(x + 3).This is similar to the graph of log2(x), but is shifted"+ 3" is not outside of the log,
the shift is not up or down First plot (1,0), test the shiftingThe log will be 0 when the argument, x + 3, is equal to 1.
When x = –2. (1, 0) the basic point is shifted to (–2, 0)So, the graph is shifted three units to the leftdraw the asymptote on the x= -3
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The graph of y = log2(x + 3) Looks like this:
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Remember:
• You may get some question about log like for example:
• Log2 )x+15( = 2
• Solution:
• 2^2= x+15x= -11, which can never be realTherefore, No Solution
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Compare between logarithm and exponential graphs:
xxf 3)(
xxf 31 log)(
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The Equations y = b x and x = log b y say the same thing.
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y = loga x
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
y =- loga x
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y = log2 (-x)
.
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
Y=loga(x+2)
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Y=loga(x+2)
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
y = loga (x-2)
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
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y = logax +2
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
y = loga x -2
1 2 3 4-1-2-3-4
x1
2
3
4
-1
-2
-3
-4
y
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