chapter 2- exponential & logarithmic function

7/28/2019 CHAPTER 2- Exponential & Logarithmic Function

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CHAPTER 2- Exponential &

Logarithmic Function



Exponential Function

An exponential function has a “constant” which is raised to a variablepower, like y=2 x the basic shape of which when drawn, indicates a rapidly

increasing function.

Recall that a table of values consists of an ordered pairs of numbers (x,y)

representing a function or describing points in a coordinate plane. Consider

the rule y=2 x for the geometric progression 2,4,8,16,….

domin: x/x is R number

range: x/x is all positive R numbers

examples:

y=2 x

x -3 -2 -1 0 1 2 3 4

y 1/8 1/4 1/2 1 2 4 8 16



b0=1

20=1

b-1= b ≠ 0

bm/n= bm

Any function raise to 0 is one

1 bm

n



Theorem 2.1

if b > 0, b ≠ 1 & if y > 0, there existence one & only onereal number x such that

y= bx

Theorem 2.2

if b > 0, b ≠ 1 & if x, & x2 are two numbers such that bx1

=bx2 when x1 =x2



25x=6 5x =? 36x+1=5 62x+2 =?

52x=6 54x= 36 62(x+1)=5 62x+2= 5

52(2x) =6 62x+2 =5

54x =36

63(x+2)= 53 2162x+2 = ?

66x+6= 125 63(x+2)=?

63(x+2)= 125



Seatwork

Tell which of the following define an

exponential

1) y=-3x

2) y=3x

3) y=x2

+14) y=(5) -x

5) y=3x2



Homework

Graph the following exponential function

1) y=3x

2) y=4x

3) y=5x

4) y=6x 5) y=7x



Logarithmic Function

The logarithm of a number is the exponent by

which another fixed value, the base, must be raised to

produce that number. For example, the logarithm of

1000 to base 10 is 3, because 1000 is 10 to the power

3: 1000 = 10 × 10 × 10 = 103. More generally, if x = by ,then y is the logarithm of x to base b, and is written y =

logb( x ), so log10(1000) = 3.



y=logb x x=by

y=log3 x x=3y

x 1/3 1 3

y -1 0 1

y

x



1) 0 function x=1 logb 1=0

2) 1 function x=b log3 3=1

3) X > 1 positive

4) X < 1 negative

5) Increasing fx

1) loga (AB)= loga A+B+loga





Log3 (9) (27) = log3 9+ log3 27=5 y=log5 1/125 3x =1/125

3x = 9 3x = 27

3x = 32 3x = 33

x= 2 x= 3

Log3 63-log3 7=2 log10 (3 √ 125) (0.0356) 2 3

Log3 3(63/7) (36.3)(√5.42)

Log3 9 3 log10 3√ 125+ 2log10 .0356

3x

=9 log10 36.2- log10 √ 5.423x = 32 1/3 log10 125+ 2log10 0.0356

x= 2 1/ 3 log10 36.2-1/2log10 5.42

log10 125+ 6log 0.0356

3 log10 36.2-3/2log10 5.42



Seatwork

Convert log to exponential

1) log4 16=2

2) log3 18=4

3) log2 1/32=-54) logb3=2

5) log1/2 2=8



Common Logarithmic

The common logarithm is the logarithm with base 10. It’scalled the common logarithm because we use abase-10

number system in our daily lives. This type of logarithm

is often simply written as log(x). log(x) gives you the

number y such that 10y = x.

The common logarithm has all the same usefulproperties

as any other logarithm:

1) log(xy) = log(x) + log(y)

2) log(x/y) = log(x) – log(y)

3) log(xy) = y * log(x)

4) log 0.00635= -2.2041

5) log 6.25= 0.7959



Finding the common logarithm log M

Log M= log N+log10k

= log N+k

= k+log N=k+M



Seatwork

1)What is log(1000)?

2)What is log(0.1)?

3) What is log(1)?4) What is log(25) + log(4)?

5) What is log(50) – log(5)?



Homework

1)Compute 4√ 3.528 using log

2)Compute 3√ 326 (0.00253)

(2.03)2

3)Compute 2√ 4.15 3using log

4)Compute 5√ 2.122using log

5)Compute 3√ 2.105 using log



Natural Logarithmic

The natural logarithm is the logarithm to the base e,

where e is an irrational and transcendental constant

approximately equal to 2.718281828. The natural

logarithm is generally written as ln x , loge x or

sometimes, if the base of e is implicit, as simply log x .

Parentheses are sometimes added for clarity, giving

ln( x ), loge( x ) or log( x ). This is done in particular when the

argument to the logarithm is not a single symbol, in order

to prevent ambiguity



e y =M then y= ln M

e y =21 y=21

y= 3.0445Theorem 2.3

if a & b are bases & m > 0

Logb M= logaMloga b



Find the value of loge 10

Log10 = 1 = 2.203

Loge 0.434Find the value of loge 8

Log10 = 0.9031 = 1.893

Loge 0.4771



Seatwork

Find the value of each of the following.

1) log4 48

2) loge 573) ln 1/3.6

4) ln 1/0.089

5) log2 6



Homework

Find approximately the values of y

1) y= log8 42

2) y= loge 3523) log7 y=3

4) loge y=2.42

5) ln y=1.50



Solution of Exponential &

Logarithmic EquationIn previous topic we learned about the exponential

and logarithmic functions, studied some of their

properties, and learned some of their applications. In this

topic we show how to solve some simple

equations which contain the unknown either as an

exponent (exponential equation) or as the argument

of a logarithmic function.



Example: 4x+1 = 3

2x = 7 log4 4x+1 = log4 3

log2 2x = log2 7 (x + 1) log4 4 = log4 3

x log2 2 = log2 7 x + 1 = log4 3

x = log2 7 2.807 x = log4 3 1



3x-3 = 5 e2x-3ex + 2 = 0

x - 3 = log3 5 (e x)2x -3e x + 2 = 0

x = log3 5 + 3 (ex

-1)(ex

- 2) = 0e x - 1 = 0

e x = 1

x = 0



Solve the equation 7x+1 =32x-3 for x

7x+1 =32x-3

(x+1) log 7= (2x-3) log 3

Xlog 7+ 3log 3= 2xlog 3 -xlog7

Log 7 +3log 3= x(2 log 3-log 7)2log 3-log 7 2log 3-log 7

X= log 7 + 3 log 3

2log 3- log 7

= 0.8451+ 1.4310.9542-0.8451

= 2.2764

.1091

=20.8653



Seatwork

1) Solve the equation 3x log 7+ xlog 5 = log 35 + log 5 for x

2) Solve the equation 72x-7x-1 = 0 for x

3) Solve the equation 3xex + x2ex = 0

4) Solve the equation log2(x + 2) = 55) Solve the equation log7(25 -x) = 3



Homework

1) Solve the equation ex +3-x =2 for x

2) Solve the equation 2x-1 +2-x =4 for x

3) Solve the equation log(3x+2)= log(x-4)+1 for x

4) Solve the equation log(2x+4)= log(x-1/2)+2 for x

5) Solve the equation log(4x-1 -1)= log (2x-2 -1)-2 for

x



END………………..

chapter 2- exponential & logarithmic function

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