THE LAWS OF LOGARITHMS

Patterns & Relations #3

Prerequisites

1. Simplify.

a) x 4 x 2 b) 105 103 c) am an

d) x 3 4 e) 105 2

f) am n

g) x 8

x 4 h) 107

103 i) am

an

2. Evaluate.

a) 271

3 b) 641

2

3. Write a1

n without exponents.

Answers to Prerequisites

1. a) x6 b) 108 c) am+n

d) x12 e) 1010 f) amn

g) x4h) 104 i) am-n

2. a) 3 b) 8

3.

an

Law of Logarithms for Powers

Read p.79 of your text One example is in base 10 The second example is in base a

The law states:

If x and n are real numbers, and x 0, then

loga xn n loga x a 0, a1

Example

3x 20

log3x log20

x log3 log20

x log20

log3

x 2.72683

You try…

Solve the equation 5x = 40 to 5 decimal places.

The solution

5x 40

log5x log 40

x log5 log 40

x log40

log5

x 2.29203

Another Example

let 9 5x

log9 log5x

log9 x log5

log9

log5x

x 1.36521

You try…

The solution

let 3x 14

log3x log14

x log3 log14

x log14

log3

x 2.40217

Law of Logarithms for Multiplication

Read p.81 of your text One example is in base 10 The second example is in base a

The law states:

If x and y are positive real numbers, then

loga xy loga x loga y a 0, a1

Example

120 1.12n 4000

log(120 1.12n ) log 4000

log120 log1.12n log4000

log120 n log1.12 log4000

n log1.12 log 4000 log120

n log4000 log120

log1.12

n 30.9

You try…

Solve the equation 15 x 1.08n = 20 000 to 1 decimal place.

The solution

15 1.08n 20000

log(15 1.08n ) log20000

log15 log1.08n log20000

log15 n log1.08 log20000

n log1.08 log20000 log15

n log20000 log15

log1.08

n 93.5

Law of Logarithms for Division

The law states:

If x and y are positive real numbers, then

loga

x

y

loga x loga y a 0, a1

Example

log64 log640

log64

640

log1

10

log10 1

1

You try…

Simplify: log550 – log5 0.4

You try…

log5 50 log5 0.4

log5

50

0.4

log5(125)

3

Textwork

p.83/ 1-19