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Exponential Functions and Logarithmic Functions
Standards 11, 13, 14, 15
Using Common Logarithms with other bases than 10
More exponential equations
Common Logarithm or base 10 logarithms
Solving Exponential Equations
Logarithmic Functions: Comparing to exponential
Exponential Functions: Introduction
Logarithmic Equations
Natural Logarithms
END SHOWPRESENTATION CREATED BY SIMON PEREZ. All rights reserved
2
STANDARD 11:
Students prove simple laws of logarithms. 11.1 Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. 11.2 Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step
STANDARD 13:
Students use the definition of logarithms to translate between logarithms in any base.
STANDARD 14
Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.
STANDARD 15:
Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is some-times true, always true, or never true.
ALGEBRA II STANDARDS THIS LESSON AIMS:
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3
ESTÁNDAR 11:
Los estudiantes prueban simples leyes de logaritmos.
11.1 Los estudiantes entienden la relación inversa entre exponentes y logaritmos y usan esta relación para resolver problemas que involucran logaritmos y exponentes.
11.2 Los estudiantes juzgan la validez de un argumento de acuerdo a si las propiedades de los números reales, exponentes y logaritmos han sido correctamente aplicados en cada paso.
ESTÁNDAR 13:
Los estudiantes usan la definición de logaritmos para convertir de una base a otra base logaritmos.
ESTÁNDAR 14:
Los estudiantes entienden y usan las propiedades de logaritmos para simplificar expresiones numéricas logarítmicas e identificar sus valores aproximados apropiados.
ESTÁNDAR 15:
Los estudiantes determinan si un estatuto algebraico que involucra expresiones racionales, expresiones radicales, o logaritmos o funciones exponenciales es algunas veces cierto, siempre cierto o nunca verdadero.
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4
Standards 11, 13, 14, 15
Exponential Functions:
42 6-2-4-6
2
4
6
-2
-4
-6
8 10-8-10
8
-8
10
x
yy= 2
x
DEFINITION OF EXPONENTIAL FUNCTION:
An equation of the form , where a = 0, b > 0, and b = 1, is called an exponential function with base b.
y = a bx
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5
Standards 11, 13, 14, 15
Simplify the following expressions:
32
32
= 3322
= 32 32
= 364
= 38
= 6561
4(5 )(5 )2 3 2 3- 4(5 )
2 3 + 2 3-=
315 3
313 3
315 3
= 315 3 13 3-
= 32 3
4=
4(5) =0
1
= (3)23
= 93
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6
Standards 11, 13, 14, 15
Find the value a if the graph of an exponential function of the
form passes through the given point:
A(3, 45) y = a 4x
y = a 4x
45 = a 43
45 = a 6464 64
a= 45 64
B(2, 64) y = a 4x
64 = a 42
64 = a 1616 16
a= 4
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7
Standards 11, 13, 14, 15
Solve the following exponential equation or inequality:
16 42x + 1 2x + 12
=
2x + 1 2x + 12=24 22
2 24(2x + 1) 2(2x + 12)
=
4(2x+1) = 2(2x + 12)
8x + 4 = 4x + 24-4 -4
8x = 4x + 20
-4x -4x
4x = 204 4
x = 5
27 94x - 2 3x + 9
<
4x - 2 3x +9<33 32
3 33(4x - 2) 2(3x + 9)
<
3(4x- 2) < 2(3x + 9)
12x - 6< 6x + 18+6 +6
12x < 6x + 24
-6x -6x
6x <246 6
x < 4
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Standards 11, 13, 14, 15
x 2 = y y
-1 2 = y
2 2 = y
3 2 = y
6 2 = y
y 2 = x x
2 = 0.5 0.5
2 = 4 4
2 = 8 8
2 = 64 64
x
-1
3
2
6
y
y
y
y
0.5
4
648
-1
2
36
y
2 = yx
2 = xy
Getting the INVERSE
for the exponential function:
Log x = y2
Solving for y:
y=x
x
y y= 2x
Log x = y2(0,1)
(1,0)
LOGARITHMIC FUNCTIONS
So the logarithmic functions and exponential functions are inverse one from the other!
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9
Standards 11, 13, 14, 15
n = bp
p = log nb
Exponential Equation Logarithmic Equation
number
81 = 34 4 = log 813
125 = 53 3 = log 1255
279936 = 67 7 = log 2799366
DEFINITION OF LOGARITHM
Suppose b > 0 and b = 1. For n > 0, there is a number p such that log n=p if and only if b = n.b
p
base
exponent or logarithm
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10
Standards 11, 13, 14, 15Solve the following logarithmic equations:
x = log 2433
x = log 2433
3 = 243x
3 = 3x 5
x = 5
5 = log 7776b
5 = log 7776b
b = 77765
b = 65 5
b = 6
4 = log n2
4 = log n2
n = 24
n = 16
Log (6x + 2) = log (3x +8)6 6
6x + 2 = 3x + 8-2 -26x = 3x + 6
-3x -3x
3x = 63 3
x = 2
Suppose b > 0 and b=1. Then log x = log x if and only if x = x
b b1 2
1 2
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11
Standards 11, 13, 14, 15
Evaluate each expression:
log 1282 = log 227 log b = xb
x
= 7
8 log (16)8 = 16 b
log xb = x
y=x
x
y y= bx
Log x = yb(0,1)
(1,0)
Remember that exponential and logarithmic functions are mutually inverse!
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12
Standards 11, 13, 14, 15
4log 3 + 2log 5 = log x7 7 7
log 3 + log 5 = log x7 7 74 2
log 81 + log 25 = log x7 7 7
log (81)(25) = log x7 7
log (2025) = log x7 7
x = 2025
with b=1log m = p log mbp
b
Suppose b > 0 and b=1. Then log x = log x if and only if x = x
b b1 2
1 2
log mn = log m + log nb b b with b=1
Solve 4log 3 + 2log 5 = log x:7 7 7
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13
Standards 11, 13, 14, 15
6log 2 - 2log 4 = log x5 5 5
log 2 - log 4 = log x5 5 56 2
log 64 - log 16 = log x5 5 5
log (4) = log x5 5
x = 4
with b=1log m = p log mbp
b
Suppose b > 0 and b=1. Then log x = log x if and only if x = x
b b1 2
1 2
Solve 6log 2 - 2log 4 = log x:5 5 5
log = log x5 56416 log = log m - log nb b b with b=1 m
n
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14
Standards 11, 13, 14, 15
n =10p
p = log n10
number
base
exponent or logarithm
Exponential Equation Logarithmic Equation
COMMON LOGARITHM: LOGARITHM WITH BASE 10
p = log n
Most Calculators only have COMMON LOGARITHM or Logarithm with base 10!
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15
Standards 11, 13, 14, 15
Find the value for each logarithm and state the characteristic and mantissa:
Log 0.0008 = -3.09691
-3.09691 + 10 - 10
6.90309 - 10 6-10= -4
6.90309 -6=0.90309
Characteristic
Mantissa
To find the characteristic and mantissa of a negative logarithm, it is necessary to express the exponent as a sum of an integer and a positive decimal
Log 69.8 = 1.84386
1 Characteristic
0.84386 Mantissa
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16
Standards 11, 13, 14, 15
Calculating logarithms with a base other than 10 using Common Logarithm:
log n =alog nLog a
b
bif b=10 then log n =a
log nLog a
log 1302
log 130
log 2=
=2.113940.30103
= 7.02237
log 2106
log 210
log 6=
=2.322220.77815
= 2.98428
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17
Standards 11, 13, 14, 15Solve the following exponential equation:
57x -3
84x + 5
=
log 8 = log 57x -34x + 5
(4x + 5)log 8 = (7x-3)log5
4x log 8 + 5 log 8 = 7x log 5 -3 log 5
-5 log 8 -5 log 8
4x log 8 = 7x log 5 -3 log 5 - 5 log 8
-7x log 5 -7x log 5
4x log 8 – 7x log 5 = -3 log 5 – 5 log 8
x(4log 8 – 7 log 5) = -3 log 5 – 5 log84 log 8 – 7 log 5 4 log 8 – 7 log 5
x =-3(.69897 )- 5(.90309) 4(.90309) - 7(.69897) x= 5.16
with b=1log m = p log mbp
b
This method is useful when the base of the exponential expressions can’t be equal!
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18
Standards 11, 13, 14, 15
n =ep
p = log n e
number
base
exponent or logarithm
Exponential Equation Logarithmic Equation
NATURAL LOGARITHM: LOGARITHM WITH BASE e
p = LN
e= 2.718281828459
It is important to observe that the Exponential Function and the Natural Logarithm Functions are inverses one from the other.
NATURAL LOGARITHM
Most calculators have them as:
e x and LN
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