unit 10: exponential and logarithmic functions. - matematicasmiguel71

Unit 10: Exponential and Logarithmic Functions. Mathematics 4th E.S.O.

Teacher: Miguel A. Hernández

UNIT 10: EXPONENTIAL AND LOGARITHMIC FUNCTIONS. Exponential Functions: The functions be a positive real number, a≠1 . Examples: Graph the functions

• • • •

y=2

x

y=a

and

x

are called exponential functions. The base a can x



1 y= 2

.

The domain of the function y=a x is ℝ . The function y=a x is continuous in ℝ . The graph of y=a x passes through the points (0,1) and (1,a). If a1 , the function is increasing. If a1 , the function is decreasing.

Example: Match each graphs with its equation: I)

y=3 x

II)

y=1,5 x

III)

y=0,4 x

1

IV)

y=0,7 x



Other types of exponential functions: Functions of type The functions k x y= a  .

y=a kx :

y=a kx are also exponential functions. Their graphs are similar to the graph of

Examples: The function The function

y=22x can be considered as the function y=2

Functions of type

x 2

can be considered as the function

x

y=22x= 2 2  =4 x . x 2

1 x 2

  =  2 

y=2 = 2

.

y=a x b :

The graph of the functions y=a x b are the graph of the functions up if b is positive, and b units down if b is negative. Example: Represent the graph of the functions:

Functions of type

x

y=2x 1 and

y=a x translating b units

x y=2 −3 .

y=a xb :

The graph of the functions y=a xb are the graph of the functions the left is b is positive, and b units to the right if b is negative. Example: Represent the graph of the functions:

y=2x1 and

2

y=a x translating b units to

y=2x−3 .



Logarithms: In a simplest form, a logarithm answer the question: How many times do we multiply a number to get another number? Example: How many times do we multiply 2 to get 8? 2·2·2=8 , so we need to multiply the number 2 three times to get 8. In other words, what is the “x” number such that 2 x =8 ? x

2 =8 ⇔ x =3

“3” is the logarithm of 8 respect to base 2, and we write this as log 2 8=3 The logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number. x

a ∈ℝ , a0 , a≠1 .

log a y= x ⇔ a = y

Examples: Calculate: a) log 2 4

b) log 2 16

d) log 2 2

e) log 2

g) log 3 3

h) log 3 9

i) log 3 27

j) log 3 81

k) log 3 1

l) log 3

Obviously:

log a a=1 ,

c) log 2 1

 1 2

f) log 2

 1 4

 1 9

x log a a = x .

Sometimes you will see a logarithm without a base, like this: log 100 . This means that the base is really 10. It is called a “common logarithm”. On a calculator is the “log” key. Example: log 100=log 10=1000=3 . Another base that is often used is e (“Euler's number”) which is approximately 2,71828 … (an irrational number). This is called a “natural logarithm”. On a calculator is the “ln” key. Example: ln 7,389=log e ≃2 . 3



Properties of logarithms: log a 1=0 and log a a=1 .

1.

2. Product rule: log a  x · y =log a x log a y . 3. Quotient rule: log a



x =log a x−log a y . y

4. Power rule: log a x n=n · log a x . 5. Change of base rule: log a x=

log b x . log b a

Your Turn 1. Calculate the following logarithms: 1 8

1 10

a) log 3 243

b) log 2

e) log 2  2

f) log a a5

g) log a  a 2

h) log a

i) log 5 1

j) log 2 0,0625

k) log 5 0,04

l) log  2 2

c) log

d) log 0,0001

3

  1 a5

2. Find the base of the following logarithms: a) log b 10000=4

b) log b 125=3

c) log b 4=−1

d) log b 3=

3. Expand each logarithm: x y6 c) log 3 x · y · z a) log

b) log  a · b 

2

d) log x · y · z 2  4

1 2



4. Condense each expression to a simple logarithm: b) 6log 3 u6log e v

a) ln x−4ln y c)

2log 7 3

d) 4log b−log c

5. Knowing log 5≃0,6989 , calculate using the properties of logarithms: a) log 50

b) log 0,05

c) log 25

d) log  5

e) log 2500

f) log 125

g) log

 1 5

h) log 5 10

6. Knowing log 3≃0,4771 , calculate using the properties of logarithms: a) log 0,3

b) log 3000

c) log 81

d) log

e) log 4 3

f) log 0,09

 1 9

7. Use the change of base rule to calculate the logarithms: a) log 2 0,5

b) log 5 120

c) log 3 0.24

d) log 0,5 2

Logarithmic function: The function y=log a x are called logarithmic functions. The base “a” can be any positive real number a≠1 . Look at the graphs of

y=2x and

y=log 2 x .

5



If the point (-3,1/8) lies on the graph of y=log 2 x ,

y=2x , then the point (1/8,-3) lies on the graph of

If the point (-2,1/4) lies on the graph of y=log 2 x , … and so on.

y=2x , then the point (1/4,-2) lies on the graph of

In general, if we have two functions, f(x) and g(x), where if (a,b) lies on the graph of f(x), then the point (b,a) lies on the graph of g(x), we say that f is the inverse function of g and vice versa. The inverse function of f is denoted f −1 (read f inverse, not to be confused with exponentiation).

The graphs of the functions y=2x and y=log 2 x are symmetric with respect the line y=x (identity function). In general, graphs of inverse functions, f and f −1 , are symmetric with respect to the line y=x . The domain of f is the range of f −1 , and vice versa, the range of f is the domain of f −1 .

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Look at the graphs of

• • • •

y=log 2 x and


y=log 1 x . 2

The domain of the function y=log a x is 0,∞ . The function y=log a x is continuous in its domain. The graph of the function y=log a x passes through the point (1,0) and (1,a). If a1 , the function is increasing. If a1 , the function is decreasing.

Example: Graph the functions

y=log 3 x and

y=log 1 x . 3

Keywords: Exponential function=función exponencial (función polinómica de primer grado) logarithm=logaritmo common logarithm = logaritmo decimal natural logarithm= logaritmo neperiano Logarithmic function= función logarítmica inverse function= función inversa

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