(chapter 4) Exponential & Logarithmic Function

QQM 1023 Managerial Mathematics

5.1 INTRODUCTION TO EXPONENTIAL FUNCTION An exponential function involves a constant (base) raised to the power of a variable (exponent) x, t etc.

DEFINITION 5.1.1(a): Exponential function with base a The function

f

is defined by

y = f (x) = a x a > 0 , a ≠ 1 and the exponent x exponential function with base a. where

is any real number Æ

DEFINITION 5.1.1(b): Exponential function with base e (2.71828...) The function

f

is defined by

y = f (x ) = e x e ≅ 2.71828... , and the exponent x is any real numbers Æ exponential function with base e. or natural where

exponential function.

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Example 1 Determine whether the given functions are an exponential function or not. If it is, then determine the base for each function.

a) y

= 2x

b)

x/3 d) y = (11)

e)

g (t ) = 3−t

y = e −0.05 x

c)

y = t −3 t

f)

h( x ) = (0.5)

g)

g ( x) = 2.7186 x

5.2

SKETCHING AN EXPONENTIAL GRAPH.

y = a x and a ≠ 1 for − ∞ < x < ∞ . Therefore the graph for

Let say

y = ax a)

If

1.2 x

can be sketch by replacing several values for x.

a = 2 . Then the exponential function would be y = 2 x

Build a table with several values of x, and find the corresponding values for y:

x

-2

-1

0

1

2

y = 2x Based on the table, we got several points that resides on the graph. Therefore, we just plot all the points and draw a curve that connects all those points.

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4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 -3

b)

-2

Now let say

1 y=   2

-1

a=

0

1

2

3

1 . The exponential function would be 2

x

Try to sketch the graph for this function. i. Build a table:

-2

x

1 y=  2

-1

0

1

2

x

ii. Based on the table, plot all of the points and draw a curve:

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4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 -3

c)

-2

-1

0

1

If the exponential function has the base

2

3

e. Where y = l x ,

Build a table:

x

-2

-1

0

1

2

y = ex Therefore the graph would be: 8 7 6 5 4 3 2 1 0 -3

-2

-1

0

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1

2

3

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d)

For the exponential function

x

-2

y = l− x

-1

0

1

2

y = e− x And the graph would be: 8 7 6 5 4 3 2 1 0 -3

-2

-1

0

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2

3

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5.3 PROPERTIES OF EXPONENTIAL FUNCTION If

a

b is any positive real numbers and x, y

and

is any rational

number, therefore;

Property 1: a

x y

a =a

Example 2:

a)

32x . 34

x+ y

Property 2:

 7 3x  5 7

a)

b) 3

b) 5-x . 5x =

ay

= a x− y

Example 3:

Simplify

=

ax

  = 

2x+2

3

Simplify

=

x-1 2

2 c)   5

3p

r

2 •  = 5

Property 3:

(a x )y = a xy

(5 )

3x 3

(

2 x +3 b) 2

Property 5:

Property 4:

(ab )x = a xb x Simplify

Example 5:

=

a)

((4)(5))−t

=

=

b)

((7 )(3))

=

)

x

x ax a   = x b b

Example 6 : 3 a)   2

c)

Simplify

Example 4: a)

3k = 39

x

Property 6:

Simplify

Example 7:

y

a− x =

1 ax

Simplify

−5 t

=

a) 5

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- (x+1)

=

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Property 7:

y

ax / y = ax Simplify

Example 8:

Property 8:

Example 9:

a) 9 3/2 =

a) -20 =

b) 6.5 ½ =

b) 23450 =

c) 12 -2/3 =

c) e 0 =

Property 9:

a0 = 1 Simplify

a1 = a

Example 10: a) 161 = b) 2.7181 =

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5.4 EXPONENTIAL FUCTION vs LOGARITHMIC FUNCTION The exponential function describe a value that is obtained by raising a constant to the power of a variable/unknown. For example:

y = 2x,

When the input x = 3 , therefore the output Æ y

= 23 = 8

Here, the variable y is a dependant variable (output), meanwhile x is an independant variable(input). Alternatively, the above equation can be switch by making x as the dependant variable (output) and y as the independant variable (input). This equation is known as the logarithmic function. Therefore, the above equation; equation;

y = 2x

can be change into a logartihmic

x = log 2 y

Here, if the input

y=8

therefore, output

of what (x) equal to 8) --> therefore; x

x = log 2 8 (2 to the power

= 3

CONCLUSION: EXPONENT

x

y=a

LOGARITHMIC

Can be written as

where a > 0 and a ≠ 1

x = log a y where a > 0 and a ≠ 1

Example 11: Rewrite the following equations in a logarithmic form.

a) 49 = 72

b) 25 = 32

c) y = 7x

d) 0.5 -3q = p

e) 42a = b Chapter 5: Exponential & Logarithmic Function

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Example 12:

Rewrite the following equations in an exponential form.

a) 1 = log 27 3 3

b) log 2 32 = 5

b) log a 4 = b

d) log 4 64 = p

5.5 INTRODUCTION TO LOGARITHMIC FUNCTION The logarithmic function with base a, where a>0 and a ≠ 1is denoted by loga and defined by;

x = log a y If and only if

ax = y.

For example: i)

y = log 2 x (if a = 2)

ii)

y = log 5 x (if a = 5)

DEFINITION 5.2.1(a): Logarithmic Function with base 10 Logarithms to the base 10 are called common logarithms . The subscript 10 is usually omitted from the notation. For example: i)

y = log10 x can be written as y = log x .

ii)

However,

y = log3 x

cannot be written as

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y = log x 123

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DEFINITION 5.2.1(b): Logarithms to the base

Logarithmic Function with base e

e. are called natural logarithms. We use notation

“ln” for such logarithms. For example:

y = ln x

i)

means

y = log e x

5.6 SKETCHING A LOGARITHMIC GRAPH Let say a logarithmic function

y = log x and x ≠ 0 . Therefire the graph

for this function can be sketch by: Form a table:

x

0.5

2

4

6

8

10

y = log x

-0.3010

0.3010

0.6021

0.7781

0.9031

1

Sketch the graph: 1.2 1 0.8 0.6 0.4 0.2 0 -0.2

0

2

4

6

8

10

12

-0.4

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Example 13: Fill in the blanks and then sketch the graph for y = ln (4x+1) x

0.5

1

2

3

4

ln (4x + 1)

5.7 PROPERTIES OF LOGARITHMIC FUNCTION

PROPERTY 1:

logb (MN ) = logb M + logb N

Example 14: a) log 10 (xy) = b) log 5 (ab2) = c) log e (2πj) =

M  logb   = logb M − logb N N

PROPERTY 2 :

Example 15: a) log 10 (a/b)= b)

c)

log1/ 2

5 = x

2x +1 ln 3 =

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logb M r = r logb M

PROPERTY 3 :

Example 16: a) log 2 4t = b) ln p

q −1

=

PROPERTY 4 :

 1 logb  M

  = − logb M 

Example 17: a)

 1  log10  = 2.5  

b)

1 ln   = π 

PROPERTY 5 :

( )

logb 1 = logb b0 = 0

Example 18: a)

log 3 1 =

b)

ln1 =

c)

logπ 1 =

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logb b = 1

PROPERTY 6:

Example 19: a)

log 3 9 =

b)

log 0.5

c)

ln e =

1 = 2

logb b r = r

PROPERTY 7:

Example 20: 2

a)

log 3 9 =

b)

ln eπ

PROPERTY 8 :

blogb M = M

Example 21: a) b)

7 log7 x =

e ln( 2 x −1) =

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log b M =

PROPERTY 9 :

log M ln M atau ln b log b

Example 22: a)

log 3 45 =

b)

logπ e =

5.8 Solving an exponential equation Steps: 1. Rewrite the exponential equation in a form of logarithmic equation. (or simply put “log” on both side of the equation) 2. Used the logarithmic properties to solve the equation.

3.

Example 23

Solve the following expopnential equations:

a)

2x = 8

b)

3 x = 27

c)

25 x = 5

d)

163 x = 2

e)

94 x = 3

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Example 24:

Solve the following equation to find the value of x;

a)

36 x = 2(32x )

b)

4 16 x = (52x ) 5

c)

43 x = 8 x

d)

49 x = 7 x +1

Example 25: x2

− 94−x = 0

a)

3

b)

2 x − 82 − 4 x = 0 2

c)

5

x2

− 25 x − 2 = 0

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5.9

Solving a logarithmic equation

Steps: 1. Rewrite the logarithmic equation in a form of exponential equation. 2. Solve the equation using the exponential properties.

Example 26:

a)

x log3 x = 3

b)

ln l3 = x

c)

l4 x = 2

d)

log 4 x = −3

e)

lln 2 x = 5

f)

log(3 x − 1) − log( x − 3) = 2

g)

log 2 (2 x − 3) = 2 − log 2 (x + 1)

h)

ln x 4 + ln x = 24 ln l

Find the value for

that satisfy the following equations:

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5.10 APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTION 5.5.1

FINANCE

Assume that we invest sum amount of money RM P, for t years. The investment earned r% of interest which is compounded m times a year. Therefore, the total amount of the investment (S) after t years can be calculate using the formula:

mt

 r S = P1 +   m Where;

S=

Value of the investment after t years (together with the interest) – Compound amount.

P= r= m= t=

Initial amount that was invested – Principal amount interest rate number of conversion investment period (number of years)

For m (number of conversion): •

Annually = once a year Æ m = 1

•

Semiannually = twice a year Æ m = 2

•

Quarterly = 4 times a year Æ m = 4

•

Monthly = 12 times a year Æ m = 12

Meanwhile, the amount of interest received (compound interset, Im) is obtain by:

Im = S − P

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Example 27: Find the value of an investment, S and the compound interest earned, Im for the principal investment amount of RM 1000 with 6% of interest a year : a) For investment period of 10 years and the interest is compounded semi-annually.

b) For investment period of 10 years and the interest is compounded quarterly.

Example 28: Suppose RM 5000 is invested at the rate of 9% compounded annually. Find the value of the investment after 5 years.

Example 29: Based on your answer in example 19, what is the value of the investment if the interest is compounded three times a year .

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QQM 1023 Managerial Mathematics 5.5.2 OTHER APPLICATIONS

a) Population Growth: Suppose that the population of a town at a certain period of time is P0 and it is increasing at the rate of r% per year. Therefore the population of the town after t years is given by:

P = Poert Where

P = Amount of Population after t years Po = Initial population r = rate of population growth t = period of times

Example 30: The projected population

P(t ) for a small town is given by:

P(t ) = 100000e0.05t with i)

t

is the number of year/s after 1980. What is the value for t if the population is predicted for the year 2002

ii)

Using your answer in (i), predict the population for 2002.

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Example 31: The population for Smallville at the year 1995 are 100 000. After 10 years, the population has grown to 150 000. What is the rate of the population growth for Smallville?

Example 32:

(Radioactive Decay)

A radiioactive element decays such that after t days the number of

−0.062 t N = 100 e . miligrams present is given by: i)

How many miligrams are initially present?

ii)

How many miligrams are present after 10 days?

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