1 Exponential Functions and Logarithmic Functions Standards 11, 13, 14, 15 Using Common Logarithms...

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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

http://www.mrperezonlinemathtutor.com/

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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:

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved


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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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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



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



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



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



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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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



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



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!



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


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



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


b


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



14

Standards 11, 13, 14, 15

n =10p

p = log n10

number

base



COMMON LOGARITHM: LOGARITHM WITH BASE 10

p = log n

Most Calculators only have COMMON LOGARITHM or Logarithm with base 10!


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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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


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


b

This method is useful when the base of the exponential expressions can’t be equal!


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Standards 11, 13, 14, 15

n =ep

p = log n e

number

base



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


1 Exponential Functions and Logarithmic Functions Standards 11, 13, 14, 15 Using Common Logarithms...

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