Numeracy - Southern Cross University

Numeracy

Introduction to Logarithms

Logarithms are commonly credited to a Scottish mathematician named John Napier who constructed a table of values that allowed multiplications to be performed by addition of the values from the table.

Logarithms are used in many situations such as:

(a) Logarithmic Scales: the most common example of these are pH, sound and earthquake intensity.

(b) Logarithm Laws are used in Psychology, Music and other fields of study

(c) In Mathematical Modelling: Logarithms can be used to assist in determining the equation between variables.

Logarithms were used by most high-school students for calculations prior to scientific calculators being used. This involved using a mathematical table book containing logarithms. Slide rules were also used prior to the introduction of scientific calculators. The design of this device was based on a Logarithmic scale rather than a linear scale.

There is a strong link between numbers written in exponential form and logarithms, so before starting Logs, let's review some concepts of exponents (Indices) and exponent rules.

The Language of Exponents

The power an can be written in expanded form as: an = a ? a ? a ? a ? a............? a [for n factors]

The power an consists of a base a and an exponent (or index) n.

Base

a n Exponent (or index) `The number of times it is repeated.'

`The number being repeated.'

Examples:

35 = 3? 3? 3? 3? 3 = 243

3^5= 243 3W5? 243

Multiplication or division of powers with the same base can be simplified using the Product and Quotient Rules.

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Page 1 [last edited on 16 November 2017]

The Product Rule

am ? an = am+n

When multiplying two powers with the same base, add the exponents.

32 ? 34 = 36 = 729 4-1 ? 45 ? 42 = 4-1+5+2 = 46 = 4096

b2 ? b7 = b2+7 = b9

The Quotient Rule

am ? an = am-n

When dividing two powers with the same base, subtract the exponents.

37 ? 32 = 37-2 = 35 = 243 42 ? 44 ? 43 = 42+4-3 = 43 = 64

e6 ? e4 = e6-4 = e2

The zero exponent rules can also be used to simplify exponents.

The Zero Exponent Rule

a0 =1

A power with a zero exponent is equal to 1.

30 = 1 1137500 = 1

e0 = 1

(4x)0 = 1

The power rule can help simplify when there is a power to a power.

The Power Rule

( )am n = am?n

When a power is raised to a power, multiply the exponents.

( )32

4

=

32?=4

3=8

6561

( )50

6

=

50?=6

5=0

1

( )b2=7 b= 2?7 b14

Two more useful Power Rules are:

(ab)m = ambm or ambm = (ab)m

a b

m

=

am bm

or

am bm

=

a m b

(3a)=3 33 ? a3

( ) 5a2 3 =53 ? a2?3 =125a6

4 a

=4

4=4 a4

256 a

The negative exponent rule is useful when a power with a negative exponent needs to be expressed with a positive exponent.

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The Negative Exponent Rule

a-m a= 1m or a1-m am

Take the reciprocal and change the sign of the exponent.

3-=2

1= 32

1 9

4-=1

1= 41

1 4

b-7

=

1 b7

4 a -3

=4 ? a3

=4a3

The fraction exponent rule establishes the link between fractional exponents and roots.

The Fractional Exponent Rule

1

an = n a

for unit fractions, or

( ) m

a n = n am or

m

na

for any fraction

1

3=2 2=3 3

2

= 83 3= 82 4

3

p4 = 4 p3

-1

b=4

1=1 b4

1 4b

Exponent Functions found on a Scientific Calculator

Function Square 32

Appearance of Key Example

d

3d= 9

Cube 23

D

2D= 8

Any exponent 35 Square root 16 Cube root 3 343

for W s or s S or S

3^5= 243 3W5? 243

s16= 4

qD343= 7 Generally qDgives S or S (although this varies between different makes and models)

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Any root 10 200

F or x

10qf200= 1.699

10q W200p 1.699

Generally qfgives F or qW gives x

(although this varies between different makes and models)

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Numeracy

Definition of a Logarithm

"When you think about Logarithms you should think about exponents (indices)"

The number 32 can be written as 25. Q: Does that mean that the Logarithm of 32 is equal to 5? A: Partially! The logarithm of 32 does equal 5 but only when a base of 2 is used.

As a Logarithm, this can be written as log2 32 = 5

We know that 216 = 63 the Log (logarithm) of 216 to the base 6 is 3 The log is the exponent (3); the exponent is 3 because the base used was 6. 216 = 63 is equivalent to writing log6 216 = 3 216 = 63 is written in exponential form log6 216 = 3 is written in logarithmic form

Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources +61 2 6626 9262 | ctl@scu.edu.au | scu.edu.au/teachinglearning

Page 1 [last edited on 16 November 2017]

A few more examples:

625 = 54 is equivalent to writing log5 625 = 4

1 000 000 = 106 is equivalent to writing log10 1 000 000 = 6

1024 = 210 is equivalent to writing log2 1024 = 10

Integral and fractional exponents are also possible

=1 8

1= 23

2-3 is equivalent to writing

log2

1 8

=

- 3

(alternatively

log2 0.125 =

-3 )

1

=5 = 25 2= 52 250.5 is equivalent to writing log25 5 = 0.5

Generalising all of the above:

1 = 70 is equivalent to writing log7 1 = 0

If a number (N) is written as a power with a base (b) and an exponent (e), such as N = be then logb N = e .

The base, b, must be a positive number (b > 0) and not equal to 1 (b 1).

The number, N, must also be positive (N > 0)

Examples:

Write the following in logarithmic form.

Exponential Form

(a)

36 = 62

(b)

128 = 27

(c)

1 = 90

(d)

1

1

3 = 92 ( 9 is the same as 92 )

(e)

2

4 = 83

(f)

a = c2

Logarithmic Form log6 36 = 2

log2 128 = 7

log9 1 = 0

log9

3

=

1 2

log8

4

=

2 3

logc a = 2

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The base is 6

The base is 6

216 = 63 is equivalent to writing log6 216 = 3

The exponent is 3

The log of the number is 3

Note: The logarithm of 1 to any base is always 0.

logb 1 = 0

Why? Remember the zero exponent rule

= b0 1= written as 1 b0

In Logarithmic Form becomes logb 1 = 0

Video `The Definition of a Log(arithm) 1'

In the next set of questions, the logarithmic form is given and is to be written in exponential form.

Write the following in exponential form.

Logarithmic Form

(a)

log5 25 = 2

(b)

log10 1000 = 3

(c)

log5 a = 3.5

(d)

logb 10 = 2

(e)

log2 9 = k

Exponential Form 25 = 52

1000 = 103 a = 53.5 10 = b2 9 = 2k

Video `The Definition of a Log(arithm) 2'

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Finding Log Values

Your scientific calculator can find the values of logs. Most scientific calculators have two logarithmic functions; Ordinary Logarithms and Natural Logarithms. Ordinary Logarithms have a base of 10. Because we use a base 10 number system, it seems straight forward that Logs with a base of 10 are used. The Log key on a scientific calculator has the appearance g. `When you are calculating the log of a number, you can assume it is with a base of ten unless it is indicated

otherwise.' In ordinary logarithms, when you find the Log of a number, you are finding the exponent when a base of 10 is used. Find the value of Log25 (meaning Log10 25) On your calculator, the sequence of keys is: g25= 1.397940009 If Log 25 = 1.397940009 this can be written in exponential form as 25=101.397940009 . Find the value of Log14500 (meaning Log1014500) On your calculator, the sequence of keys is: g14500= 4.161368002 If Log14500 = 4.161368002 this can be written in exponential form as 14500=104.161368002 .

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