It includes concepts of logarithms including properties and log tables. The methodology to find out values using log tables and anti log tables is also mentioned in a detailed manner. Moreover, questions related to logarithms are mentioned for practice.

1. LOGARITHMS
BirinderSingh,AssistantProfessor,PCTE

2. INDICES
 Any expression written as an is defined as the
variable a raised to the power of the number n
 n is called a power, an index or an exponent of a
 Example - where n is a positive whole number,
a1 = a
a2 = a  a
a3 = a  a  a
an = a  a  a  a……n times

3. INDICES SATISFY THE FOLLOWING RULES
 an = a  a  a  a……n times
e.g. 23 = 2  2  2 = 8
 Negative powers…..
a-n =
e.g. a-2 =
e.g. where a = 2
2-1 = or 2-2 =
n
a
1
2
1
a
2
1
4
1
22
1



4.  A Zero power
a0 = 1
e.g. 80 = 1
 A Fractional power
e.g.
n
aa n

1
3999 22
1

288 33
1


5. ALL INDICES SATISFY THE
FOLLOWING RULES IN MATHEMATICAL
APPLICATIONS
Rule 1 am. an = am+n
e.g. 22 . 23 = 25 = 32
e.g. 51 . 51 = 52 = 25
e.g. 51 . 50 = 51 = 5
Rule 2
nm
n
m
a
a
a 

822
2
2
2
222
2
2
1
303
0
3
123
2
3




..
..
ge
ge

6. Rule 3
(am
)n
= am.n
e.g. (23
)2
= 26
= 64
Rule 4
an
. bn
= (ab)n
e.g. 32
 42
= (34)2
= 122
= 144
Likewise,
n
b
a
nb
na






 if b0
e.g.
42
3
6
3
6 2
2
2
2








7. SIMPLIFY THE FOLLOWING USING
THE ABOVE RULES:
1) b = x1/4
 x3/4
2) b = x2
 x3/2
3) b = (x3/4
)8
4) b = yx
yx
4
32
These are practice questions for you to try at home!

8. LOGARITHMS
A Logarithm is a mirror image of an
index
If m = b
n
then logbm = n
The log of m to base b is n
If y = xn
then n = logx y
The log of y to the base x is n
e.g.
1000 = 103
then 3 = log10 1000
0.01 = 10-2
then –2 = log10 0.01

9. EVALUATE THE FOLLOWING:
1) x = log39
the log of m to base b = n then m = bn
the log of 9 to base 3 = x then
 9 = 3x
 9 = 3  3 = 32
 x = 2
2) x = log42
the log of m to base b = n then m = bn
the log of 2 to base 4 = x then
 2 = 4x
 2 = 4 = 41/2
x = 1/2

10. THE FOLLOWING RULES OF LOGS
APPLY
1) logb(x  y) = logb x + logb y
eg.   3232 101010 logloglog 
2) logb 





y
x
= logb x – logb y
eg.
23
2
3
101010 logloglog 





3) logb xm
= m. logb x
e.g. 323 10
2
10 loglog 

11. PRACTICE PROBLEMS
Q1: Prove that loga1 = 0 & logaa = 1
Q2: Write the following in the logarithmic form:
o 45 = 1024
o 15-2 = 1/225
Q3: Write the following in exponential form:
o log101000 = 3
o Log2(1/4) = – 2
Q4: Find log216, log327, logbb, log10105, log2 32, &
log5 5
3
, logrr5, log7 7
3
Q5: If logx243 = 5, find x.
BirinderSingh,AssistantProfessor,PCTE

12. PRACTICE PROBLEMS
Q6: Without using log tables, find x if
1
2
log10(11 + 4 7) = log10(2 + 𝑥)
Q7: Show that log 𝑏 𝑎 log 𝑐 𝑏 log 𝑑 𝑐 log 𝑎 𝑑 = 1
Q8: Show that
log3 1 +
1
3
+ log3 1 +
1
4
+ log3 1 +
1
5
+ …… + log3 1 +
1
3
= 4
Q9: Show that 7 log
16
15
+ 5 log
25
24
+ 3 log
81
80
= log 2
Q10: Find the value of
log 27 : log 8 ;log 125
log 6 ;log 5
Q11: Find the value of log3 729. 9;1 27;4/334
BirinderSingh,AssistantProfessor,PCTE

13. FUNDAMENTAL PROPERTIES -
LOGARITHMS
 Product Formula: loga 𝑚𝑛 = loga 𝑚 + loga 𝑛
 Quotient Formula: loga(𝑚/𝑛) = loga 𝑚 + loga 𝑛
 Power Formula: loga 𝑚 𝑛 = n loga 𝑚
 Base Change Formula: logn 𝑚 = loga 𝑚 / loga 𝑛
BirinderSingh,AssistantProfessor,PCTE

14. FROM THE ABOVE RULES, IT FOLLOWS
THAT
(1) logb 1 = 0
(since => 1 = bx
, hence x must=0)
e.g. log101=0
and therefore,
logb
 x
1
= - logb x
e.g. log10 (1
/3) = - log103
11
)

15. AND……..
(2) logb b = 1
(since => b = bx
, hence x must = 1)
e.g. log10 10 = 1
(3) logb  n
x = n
1
logb x
1
)

16. APPLICATION OF LOGARITHMS
 Useful in complicated calculations
 Useful in calculation of compound interest
 Useful in calculation of population growth
 Useful in calculation of depreciation value
BirinderSingh,AssistantProfessor,PCTE

17. LOG TABLES
 The logarithm of a number consists of two parts,
the whole part or the integral part is called the
characteristic and the decimal part is called the
mantissa.
 The characteristic can be known by inspection,
the mantissa has to be obtained from logarithmic
table

BirinderSingh,AssistantProfessor,PCTE

18. CHARACTERISTIC
 The characteristic of the
logarithm of any number
greater than 1 is positive
and is one less than the
number of digits to the left
of decimal in the given
number.
 The characteristic of the
logarithm of any number
less than 1 is negative and
is one more than the
number of zeroes to the
right of decimal in the
given number.
 If there is no zero, then
characteristic will be – 1.
BirinderSingh,AssistantProfessor,PCTE
Number Characteristic
9.40 0
81 1
234.5 2
12345 4
0.7 – 1
0.03 – 2
0.007756 – 3

19. MANTISSA
 The mantissa is the fractional part of the
logarithm of a given number and worked out
through log tables
BirinderSingh,AssistantProfessor,PCTE
Number Mantissa Logarithm
Log 8509 ……. .9299 3.9299
Log 850.9 ……. .9299 2.9299
Log 85.09 ……. .9299 1.9299
Log 8.509 ……. .9299 0.9299
Log 0.8509 ……. .9299 1.9299
Log 0.08509 ……. .9299 2.9299
Log 0.008509 ……. .9299 3.9299

20. KEY TO REMEMBER
 In the common system of logarithms, the
characteristic of the logarithm may be positive or
negative but the mantissa is always positive.
 A negative mantissa must be converted into a
positive mantissa before reference to logarithm
table.
 Ex: – 2.6278 = – 3 + (3 – 2.6278) = 3.3722
BirinderSingh,AssistantProfessor,PCTE

21. A NOTE OF CAUTION:
 All logs must be to the same base in
applying the rules and solving for values
 The most common base for logarithms are
logs to the base 10, or logs to the base e
(e = 2.718281…)
 Logs to the base e are called Natural
Logarithms
 logex = ln x
 If y = exp(x) = ex
then loge y = x or ln y = x

22. ANTILOGARITHMS (AL)
 The process of finding the antilog of a given
logarithm is just the reverse of the procedure for
finding the logarithm of a given number
 For finding the anti log of a log, only mantissa is
used for consulting the table.
BirinderSingh,AssistantProfessor,PCTE
Number Logarithms Antilogarithms
729 2.8627 729.0
72.9 1.8627 72.90
7.29 0.8627 7.290
0.729 1.8627 0.7290
0.0729 2.8627 0.07290
0.00729 3.8627 0.007290

23. PRACTICE PROBLEMS – LOGARITHM
TABLE
Q12: Find the value of 92.51 x 356.43 Ans: 32980
Q13: Find the value of 743.85/51.63 Ans: 14.41
Q14: Find the value of (6.285)5 Ans: 9806
Q15: Find the value of (395)1/7 Ans: 2.349
BirinderSingh,AssistantProfessor,PCTE

24. PRACTICE PROBLEMS – LOGARITHM
TABLE
Q16: If log 5 = 0.6990, find the number of digits in 521. Ans: 15
Q17: If log 7 = 0.8451, find the position of the first significant
figure in 7-35. Ans: 29th
Q18: Show that 1
1
20
100
> 100
Given: log 2 = 0.30103, log 3 = 0.4771213, log 7 = 0.8450980, log 10 = 1
Q19: Evaluate
2.389 𝑥 0.004679
0.00556 𝑥 52.14
using log tables. Ans: 0.03856
BirinderSingh,AssistantProfessor,PCTE

25. PRACTICE PROBLEMS – LOGARITHM
TABLE
Q20: Find the compound interest on Rs. 15000 for 6 years at 8%
p.a. Also find the simple interest. Ans: 8800, 7200
Q21: Find the compound interest on Rs. 4900 for 5 years, if
interest is payable half yearly, the rate for first three years being
7% p.a. and for remaining two years is 9% p.a. Ans: 2267
Q22: What is the present value of Rs. 20000 due in 3 years at
10% p.a. compounded annually? Ans: 15030
Q23: In what time will a sum of money double itself at 5% p.a.
compounded annually? Ans: 14.2
BirinderSingh,AssistantProfessor,PCTE

26. PRACTICE PROBLEMS – LOGARITHM
TABLE
Q24: A machine the life of which is estimated to be 12 years,
costs Rs. 17500. Calculate its scrap value at the end of its life,
depreciation being charged at 9 % p.a. Ans: 5636
Q25: A population of 1000 persons is expected to grow
exponentially over the next 10 years at 4% p.a.. How large
will be the population at the end of 10 years? Ans: 1479
Q26: If a population increases from 10 million to 30 million in
5 years, what is the annual growth rate? Ans: 24.6%
BirinderSingh,AssistantProfessor,PCTE