Exponents & Logarithms

EXPONENTS & LOGARITHMS EXPONENTS In the expression a b , a is called the base and b is called the exponent. Integer Exponents : When n is a positive integer, and x is a real number, then the n-th power of x is written as x n = x Bx Bx Bx Bx ……AA multiplied n times Example : f g4 1f f f

3

1f f f1f f f1f f f1f f f 1 f f f f f f f = A A A = 3 3 3 3 81

@ 2 = @ 2 A @ 2 A @ 2 A @ 2 A @ 2 A @ 2 = 64

`

a6 `

a`

a`

a`

a`

a`

but @2 = @ 2.2 A 2.2 A 2.2 = @ 64 6

a

Laws of exponents : B C for a and b being real numbers and m and n being integers am A an = am + n `

ab

am

= am A b

m

= a mn 1f f f f f f f f a@ m = m a

`

am

an

m af f f f f f f f m@n =a an d em m af af f f f f f f f f f f = m b b 0 a =1 1f f f f f f f f

w w w w w w w

a m = mp a

Example: x 3 A x 9 = x 3 + 9 = x 12 9 xf f f f f f f 9@5 =x = x4 5 x d e5 5 xf xf f f f f f f f f f = 5 2 2

f g0 2f f f

3

=1

a4 `

`

2x A 2x

= 2x

a@ 5 `

2x = 2 A x 5 = 32 x 5

`

a5

5

b c3

x2 = x6

5

@2

=

1f 1f f f f f f f f f f f f = 2 25 5

= 2x

a4 @ 5 `

a@ 1

1f 1f f f f f f f f f f f f f f f f f f f f f = ` a1 = 2x 2x

c4b

b

c@ 3

2 3 4 2x yf 3x yf f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f

Example : Simplify

x2 y

Solution : c4b

b

c@ 3

2 3 4 yf 3x yf 2x f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f

x2 y

b c4 1f 1f f f f f f f f f f f f f f f f f f f f f f f ff f f f f f f f f f f = 2x 2 y 3 A bf c3 A 2 3x y 4 x y

1f f f f f f f using a@ m = f m a

=2 x2

using ab

b c4b

4

c4 1f 1f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f ff f f f f f f f f f f y3 A f b c3 A 2 3 3 3 x y4 x y

`

1f 1f f f f f f f f f f f f f f f f f f f f f f f f f ff f f f f f f f f f = 16 x 8 y12 A f A f 3 12 2 27 x y x y 8 12 16 xf yf f f f f f f f f f f f f f f f f f f f f f f f f = f 5 13 27 x y 16 f f f f f f3 @1 = f x y 27

am

`

= am b

am

using ab

= am b

m af f f f f f f m@ n using f =a an

@2 @3 xf yf zf f f f f f f f f f f f f f f f f f f f f f f f f f k Example : Simplify j f 2 3 @4 x y z

Solution : i@ 3

h

@2 @3 yf zf f f f f f f f f f f f f f f f f f f f f f f f f f f j xf k 2 3 @4 x y z

c@ 3

= x 1 @ 2 A y@ 2 @ 3 A z@ 3 + 4 b

c@ 3

Using

m af f f f f f f f m@ n n =a a

= x @ 1 y@ 5 z1 b

c@ 3b

= x@ 1 b

= x 3 y15 z@ 3 =

3 15 xf yf f f f f f f f f f f f f f f f f z3

c@ 3` a@ 3

y@ 5

z

m

using a m a n = a m + n

i@ 3

h

m

`

Using ab Using a m `

am an

Using a@ m =

= am A b = a mn 1f f f f f f f f am

m

Radicals : n

We know that 4 where n is an integer means 4 multiplied n times. But if the power is a 2f f f f f 2f f rational number like f then what will 4 3 mean? To understand that we first need to 3 1f f f f f

understand 4 3 , which is a radical. If n is a positive integer then the principal n-th root of a is defined as 1f f f f f f w w w w w w n nw p a = a n = b when b = a

If n is even, a must be positive for b to be real, because as b = a , any real number b w w w w w w nw multiplied even number of times will be positive. So, if n is even and a is negative, p a w w w w w w w w w w w w w w w w 4 is not real. ( p@ 5 is not real and is a complex number. If n is even we must have a ≥ 0 and b ≥ 0 n

w w w w w w w w w w w w w w w w w w w w 3w 3w However, if n is odd, a can be either positive or negative. ( p5 and p@ 5 are both real ).

Properties of Radicals : w w w w w w w w w w w nw

w w w w w w w w w w w w wnw

n 1 A p ab = p a pb w w w w w w w w w w w w w w w w p n af f f f f f f f f f f f f f f f n a w w w w w w 2Ar =p nw b b

w w w w w w w w w w w w w w w w w w w w w w w w w

w w w w w w w

n 3 A mqp a = mnp a

w w w w w w w w w w w nw

4 A p a n = a if n is odd, ` a root n a n = |a| if n is even A Examples :

w w w w w w w w w w w 5fffff 5f f f f f w w w w w w 4 4w q x5 = p x =x4

Rational Exponents :

Rational exponent or fractional exponent has a rational number in the power like 2f f f f f

1f f f f f

8f f f f f

4 3 , 8 2 , a 3 etc. w w w w w w nw If m and n are integers , n>0 and a is a real number and p a is also a real number m f f f f f f f f

m f f f f f f f f

w w w w w w nw an = p a

cm

b

w w w w w w w w w w w w nw

a n = pa m

or

With this definition it is easy to see that the Laws of Exponents can also be applied to Rational Exponents. Example : @

64

1f f f f f 3

1f 1f f f f f f f f f f f f f f f f f f f f f f f 1f f f f = f = 3w w w w w w w w w w w= 1f f f p 4 64 64 3

w w w w w w w w w w w w w w w w w w w w a ` a2 3w @ 27 3 = p@ 27 = @ 3 = 9 2f f f f f

`

x

@

5f f f f f 3

c2

b

1f 1f f f f f f f f f f f f f f f f f f f f w w w w w w w w w w = f = 3w 5f f f x 3 qx 5 d

Example : Simplify

@

2x 4 y

c 2fffff

e3b

4f f f f f 5

8 y2

3

Solution : d d

4

@

2x y

e3b

4f f f f f 5

= 2 x 4 B3 y 3

d

@

= 2 x y 3

d

12

@

= 2 x 12 y 3

@

2

= 2 x 12 y 5

ed 4f f f f f B3 5

2f f f f f

83 y

2B

e 2f f f f f 3

efb w w w w wc2 2 B 2fffffg 3w p 8 y 3

12 f f f f f f f f f 5

efb 12 f f f f f f f f f w w w w wc2 3w p 5

8

= 2 A2 x 12 y 3

8y

c 2fffff 2 3

@

g 4f f f f f

y3

12 f f f f f f f f f 4f f f f f + 5 3

@ 36 + 20 f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f 15

= 32 x 12 y

@

16 f f f f f f f f f 15

EXPONENTIAL FUNCTION 3

Earlier we have studied exponents like 2 or a b . Here the exponents 3 and b are constants. Can we put a variable (say x ) in the exponent and express a function that way? x sin x ? Yes, we can. These are called exponential functions. Something like 4 or 2

In a function, if the independent variable is in the exponent, and the base is a real number greater than 0 and not equal to 1, it is called an exponential function. It is expressed as ` a f x = a x , where a is a real number, a>0, a ≠ 1. The domain of this exponential function (i.e. the possible values of x ) is the set of real numbers. a x is a real number for any real number x. x

Example : 2 , 3

@x

f g2x 1f f f

@ x2

etc are examples of exponential functions. 3 x 2 is an exponential function, as the independent variable x is in the exponent, and the base 2 is a real number greater than 0. ,

,2

Graphs of exponential functions :

To draw the graph of exponential functions y = f(x), we make a table consisting two columns : one for x and one for y . We take different values of x and calculate the corresponding values of y or f(x) . Then we plot the points in a graph and join them with a smooth curve. This is the graph of the given exponential function . We try to draw the graph of f x = 2 taking y = 2 and making the table -4 -3 -2 -1 0 1 2 3 x: 1f 1f 1f 1f 1 2 4 8 y: f f f f f f f f f f f f f 16 8 4 2 ` a

x

x

We try to draw the graph of f x = 3 taking y = 3 -4 -3 -2 -1 0 x: 81 27 9 3 1 y: ` a

@x

@x

We plot them in the graph and get the following graphs.

and making the table 1 2 3 1f 1f 1f f f f f f f f f f f 9 27 3

4 16

4 1f f f f f f f 81

The irrational number e is defined as : 1f f f f 1f f f f 1f f f f 1 f f f f f ' e=1+ f + f + f + f + 1 1! 2! 3! 4! f gn 1f f f f = n lim 1+ Q1 n It can be shown that e ≈ 2.71828

LOGARITHM If a m = N where N>0 and a>0,a ≠ 1 , b

c

then m = loga N m is the logarithm of N to the base a A b

c

Examples : 1 A Find log5 125 A We know, 5 = 125 [ then log5 125 = 3 A 2 A Find log5 0.04 A 1f 4f f f f f f f f f f f f f f f f f 1 f f f f f f @2 = = =5 We know, 0.04 = 100 25 5 2 [ then log5 0.04 = @ 2 3

`

a

3 A Find log5 @ 10 A We know, there is no real number x for which 5 = @ 10 ` a [ then log5 @ 10 is undefined A x

Rules of Logarithm : By using the laws of exponents mentioned above, we can prove the following rules of logarithm : b c for m>0, n>0, a>0 & a ≠ 1, b being any real number ,

1 A loga mn = loga m + loga n d e mf f f f f = loga m @ loga n 2 A loga f n `

a

3 A loga a b = b b c

4 A loga m x = x loga m When e is used as a base for logarithm :

The irrational number e as defined earlier in this chapter is used as a base in calculation logarithm. A special symbol “ln” is used for this. ln m = loge m Natural logarithm : The logarithm with base as e is called natural logarithm. So natural logarithm of 100 is ln 100 = loge 100 = 4.605 Common logarithm : The logarithm with base as 10 is called common logarithm. It is sometimes denoted by omitting the base. So, log 100 = log10 100 = 2

Example `: a ` a a log4 2 + log4 32 = log4 2 A 32 = log4 64 = 3 b log2 48 @ log2 3 = log2

` a

f

48 f f f f f f f f 3

g

Law 1

Law 2

= log2 16 = 4

Change of base : Sometimes we need to change the base of the logarithm. If the base is changed from b to a for the logarithm of x , then the relation is log xf f f f f f f f f f f f f f f f f af logb x = loga b xf log ` a ln xf f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f ef This can be used to change the base from 10 to e , log10 x = f = approx loge 10 2.3026

This is very useful to find logarithm with uncommon base which cannot be found directly in the calculator (the calculator gives direct values of natural and common logarithms only). So, to find log3 8 , we can use b c log 8f lnf 8f f f f f f f f f f f f f f f f f f f f (natural log) log3 8 = f common log or log3 8 = f ln 3 log 3

LOGARITHMIC FUNCTIONS A function of the form f x = loga x is called a logarithmic function. This is meaningful only when a>0 and a ≠ 1 A ` a

If we take f x = loga x = y, then by the properties of logarithm, x = a y = F y which is an exponential function of y. So, a logarithmic function can be seen as an inverse of an exponential function. Thus the functions y = loga x, and x = a y are equivalent. ` a

` a

Some properties of logarithmic function f x = loga x : 1. The domain of the function (i.e. the values of x), is the set of all positive real numbers. 2. The range i.e. the values of f(x), is the set of real numbers. 3. f(1) = 0 and f(a) = 1. 4. f is an increasing function when a>1 and decreasing function if 0
Below is the graph of y = log3 x (as 3 rel="nofollow">1, the value of y increases with the increase in x)

Below is the graph of y = log0.5 x (as 0.5<1, the value of y decreases with the increase in x)

Drawing graphs of logarithmic functions :

To draw the graph of logarithmic functions y = f(x), we make a table consisting two columns : one for x and one for y . We take different values of x and calculate the corresponding values of y or f(x) . Then we plot the points in a graph and join them with a smooth curve. This is the graph of the given exponential function . We try to draw the graph of f x = log2 x taking y = log2 x and making the table 1f 1f 1f 1f 1 2 4 8 x: f f f f f f f f f f f f f 16 8 4 2 -4 -3 -2 -1 0 1 2 3 y: We plot them in the graph and get the following graph. ` a

16 4