Laws of Logarithms

Further Mathematics SS 1 FIRST TERM

WEEK TWO

SS1 FURTHER MATHS FIRST TERM

LOGARITHMS

CONTENTS:

1. Laws of Logarithms
2. Change of Base of Logarithms.
3. Use of Tables (greater than one and less than one).
4. Logarithmic Equations.

SUB TOPIC: LAWS OF LOGARITHMS

In the last topic indices, we learnt that p = a^x , e. g 1000 = 10³ where 3 is called the index. We can express the same in logarithms form. Log _a ^p = x or log ₁₀ 1000 = 3

What then is logarithms?

The logarithms of a number p to base a, where a is a positive number not equal to 1 is the index to which a must be raised to give p. This shows clearly that indices and logarithms are the same.

Log₂8 = 3 because 8 = 2³

Log ₃9 = 2 because 9 = 3²

Laws of Logarithms

1. Log_a(pq) = Log_ap + Log_aq = Multiplication rule

e.g. if Log₃(6×5) = Log₃6 + Log₃5

1. Log_aa = 1 e.g Log₁₀10 = 1
2. Log_a(x/y) = Log_ax – Log_ay = Division Rule
3. Log_a(x)ⁿ = nLog_ax
4. Log_a1 = 0
5. Log_a(1/x) = Log_ax^-1 = -1Log_ax
6. If Log_by = 1 then y = b
7. If Log_ba = 1/Log_ab
8. Log_b = 1/nLog_bx
9. Log_by. Log_yb = 1 for b and y positive and not equal to 1
10. Log _by = Log _by / Log _ab

Examples:

1. Simplify Log₃⁹ + Log₃²¹ – Log₃⁷

Solution:

Log₃⁹ + Log₃²¹ – Log₃⁷

= Log₃(9×21÷7)

= Log₃(9×21/7)

= Log₃27

= Log₃3³ = 3 Log ₃3

But Log₃3 = 1

Therefore 3 x 1 =3

Log₃⁹ + Log₃²¹ – Log₃⁷ = 3

1. Solve completely for x in the equation 4Log_x5 = Log₅x

Solution:

4Log_x5 = Log₅x

4Log_x5 = 4/Log₅x since Log_yx = 1/Log_xy

therefore 4/Log₅x = Log₅x = 4 = (Log₅x)2

take square root of both sides

x = ±2

Therefore Log₅x = ±2

Hence, Log₅x = 2 or Log₅x = -2

x = 5² or 5^-2

x = 25 or 1/25

1. Solve the equation Log₄₍x² + 6x + 11) = 1/2

Solution:

Log₄₍x² + 6x + 11) = 1/2

x² + 6x + 11 = 4^1/2 = 2

x² + 6x + 11 – 2 = 0

x² + 6x + 9= 0

(x +3)(x + 3) = 0

x = -3 twice

CLASS ACTIVITIES:

1. Simplify the following

(a) Log₃27 + 2Log ₃9 (b) Log _x x9 (c) Log₅

1. Solve the following

(i) Log₁₀(x²+4) = 2 + Log₁₀x – Log₁₀20 (ii) Log₂2n – 2Log₈n = 4

SUB TOPIC CHANGE OF BASE OF LOGARITHMS

To change the base of logarithms, we follow the procedure below

Let Log_ax= y then x = _ay

Log_bx = yLog_ba

yLog_ba = Log_bx

y = Log_bx/Log_ba

Therefore Log_ax = Log_bx/Log_ba

Example:

Change the base of the following logarithms to base 10

1. Log₃81 (b) Log₅125 (c) Log_cx = d

Solution

1. Log₃81 = x

Log₃81 = Log₁₀81/ Log₁₀3

= Log₃3⁴/ Log₁₀3

= 4Log₁₀3/ Log₁₀3 = 3

Therefore Log₃81 = 4

1. Log₅125 = Log₃125/ Log₁₀5

Log₅125 = Log₁₀5³/ Log₁₀5

Log₅125 = 3Log5/ Log5

Therefore Log₅125 = 3

1. Log_cx = d

d = Log₁₀x/ Log₁₀c

therefore Log_cx = Log₁₀x/ Log₁₀c

CLASS ACTIVITICES:

1. Show that

SUB TOPIC: USE OF TABLES (GREATER THAN ONE AND LESS THAN ONE)

Generally logarithms with base 10 are universal. This is why we have the table of logarithms in base ten. The logarithm of any number has two parts. These are the characteristics and mantissa. The characteristic is the integer part.

Consider the Log₁₀530, Log₁₀53 and Log₁₀5.3.

The mantissas of all of them are the same. The difference is in the characteristics.

530 = 5.3 x 10² , 53 = 5.3 x 10¹ whereas 5.3 = 5.3 x 10⁰.

No Log

530 2.7243

53 1.7243

5.3 0.7243

This is useful to evaluate problems of multiplication and division

To check from table of logarithms, you will need to follow the examples

Example:

1. use tables to evaluate the following
2. 65.43 x 1453 (b) 86.31 x 0.6218 (c) 0.07304 ÷ 0.8931

Solution:

No Standard form Log

65.43 6.543 x 10¹ 1.815

1453 1.453 x 10³ 3.1623

4.9781

-we look up 65 under difference 3

8 1 5 6 Add the difference

• 2

8 1 5 8

14 under five difference 3

16.14 + 9 = 1623

–For multiplication we add

–Look up the antilog table

0.97 under 8 difference 1

9506 + 2 = 9508

The characteristic is used to determine the decimal point location.

Antilog of 0.9781 = 9.508, Hence 9.508 x 10⁴ = 95080

65.43 x 1453 = 95080

1. 86.31 x 0.6218

No Standard form Log

86.31 8.631 x 10¹ 1.9361

0.6218 6.218 x 10^-1 1.7937

1.7298

-note: 1 is called bar 1.

Mantissa is always positive but characteristic can be negative or positive; we put the negative sign on it. When there is no sign then, it is positive.

9360 + 1 = 9361

7931 + 6 = 7937

From Antilog table

.7298 = 5358 + 10 = 5368

• 1. x 10¹ = 53.6886.31 x 0.6218 = 53 .68

 

1. 0.07304 ÷ 0.8931

No Standard form Log

0.07304 7.304 x 10^-2 2.8635

0.8931 8.931 x 10^-1 1.9509

2.9126 subtract for division

8633 + 2 = 8635

9509 + 0 = 9509

Antilog of .9126 = 8166 + 11 = 8177

8.177 x 10^-2 = 0.08177

CLASS ACTIVITIES:

Use tables to solve the following

1. 23.82 x 142.8
2. 0.03167 x 102.8 x 0.325
3. 14.87 ÷ 2.314
4. (12.31)²
5. (33.28) ÷ 4.689

SUB TOPIC: LOGARITHMIC EQUATIONS

EXAMPLE 1: Solve for m in the equation

Solution

Equating powers

-4(2m+3) =

• 4(2m + 3) = 5m+1

-12 – 1 = 5m + 8m

-13 = -13m

-1 = m

EXAMPLE 2: Given that , solve for x and y respectively.

Solution

Applying the laws of logarithm

We can now equate terms

8 = yx – x ………………………………………………….. (i)

Also,

Applying the laws of logarithm,

We can now equate terms

4 = yx + x ………………………………………………………(ii)

Solving (i) and (ii) simultaneously, we eliminate yx.

8 = yx – x

-(4 = yx + x)

4 = -2x

X = = – 2

Put x = – 2 into (2)

4 = yx + x will become

4 = y(- 2) + (- 2)

4 = – 2y – 2

4 + 2 = – 2y

6 = – 2y

y =

x, y means – 2, – 3.

CLASS ACTIVITIES:

1. Find n if
2. If

PRACTICE EXERCISE

Objective test:

Choose the correct answer from the alternatives

1. Evaluate Log_0.258 (a) ½ (b)2/3 (c) -2/3 (d) -3/2
2. If 2Log₄2 = x + 1, find the value of x. (a) -2 (b)-1 (c) 0 (d) 1
3. Given that Log₃(x-y) = 1 and Log₃(2x +y) = 2, find the value of x. (a) 1 (b)2 (c) 3 (d) 4

Essay Questions:

Solve for x giving your answer correct to 3 S. F.

1. 2Log₁₀x + 3Log₁₀5 = 2
2. Using logarithm table, evaluate

÷ correct to 3 s.f

1. If , find x and y respectively.
2. Given that
3. Using logarithm table to evaluate

ASSIGNMENT

1. Without using tables, simplify
2. If m and n are positive real numbers such that

Find: (i) a relation between m and n which does not involve logarithms; (ii) the value of n if m = 16. (leave your answer as in surd in the simplest form).

1. If
2. Using logarithms table, evaluate giving your answer to three significant figures.
3. Use tables to evaluate 13.81×142.8

KEY WORDS

• LOGARITHM
• ANTILOGARITHM
• CHARACTERICTICS
• MANTISSA
• DIFFERENCE
• BAR[mediator_tech]