Workshop

The exponential and logarithmic functions

Topics Covered:

• Exponential function • Logarithmic function • Exponential decay • Solving equations with exponents and logarithms

by Dr.I.Namestnikova

1

Laws of Exponents For any positive a

an bm = am+n (an )m = an×m n a 1 n−m = a a−n = n , m a a √ √ 1 m n n n n a = a, a = am a0 = 1

2

Integer exponents:

45 = 4 × 4 × 4 × 4 × 4 {z } | 5

times

50 = 1 1 1 1 1 9−3 = 3 = × × 9 9 9} |9 {z 3

times

1 3

√ 3

Rational exponents: 2

1

• 8 3 = (8 3 )2 = 22 = 4 or 2 3

2

• 8− 3 or

2

1 3

8 = (8 ) = 64 = 64 = 4 1 1 = (8 3 )−2 = 2−2 = 4   13 1 1 2 1 8− 3 = (8−2 ) 3 = = 64 4

Example:

 3 −4 What is 5− 2 ?

Solution:

Example: What is



Solution:


3 1 −2 2

−4

−3 2

5

−2

12

= 5 2 = 56 = 15625

?

 − 32 !−2 1 2

3

=

 3 1 2

=

1 8

Exponential Functions f HxL

The letter e stands for

4

the exponential constant which is

3

approximately 2.71828. Graph f (x)

= exp(x) exp(x) = ex

2

1

-2

0

-1

1

showing exponential growth

x

2

f HxL 4

3

Graph f (x) 2

= exp(−x)

showing exponential decay

1

-2

0

-1

1

2

x

f HxL 4

3x 0.5x

2

2

-2

3x -1

f (x) = 0.5x , f (x) = 2x f (x) = 3x

x

0.5x

1

2x

Graphs

3

0

1

2

x

4

For any two functions

f1 (x) = ax with a > 0 and f2 (x) = bx with b > 0 we have

If

a > 1, b > 1 and a < b then ax < bx

If

a < 1, b < 1 and a < b then ax > bx

For any function

f (x) = ax with a > 0 we have

If

a > 1 and x < y then ax < ay

If

a < 1 and x < y then ax > ay

5

Logarithmic Functions logarithm

number

x = logaHyL base y = ax ⇐⇒ x = loga (y) 23 = 8 ⇐⇒ 3 = log2 (8)

5−2 = 0.04 ⇐⇒ −2 = log5 (0.04)

103 = 1000 ⇐⇒ 3 = log10 (1000) 5 = log4 (32) 45/2 = 32 ⇐⇒ 2

f HxL 3

ex

e are called natural logarithms and denoted ln. ln(x) = loge (x), x > 0 Logarithms to base

2

1

lnHxL -3

-2

1

-1

2

3

x

ex defined for any real x, i.e. −∞ < x < ∞

Function

-1

-2

-3

6

Logarithmic Functions f HxL 0.2

0.4

0.6

0.8

1.0

x

1.2

-0.5

Graph f (x)

log HxL 5

= log5 x,

x>0

-1.0

-1.5

-2.0

f HxL -1.2

-1.0

-0.8

-0.6

-0.4

x

-0.2

log H-xL

-0.5

Graph f (x)

5

= log5 (−x),

x