Author: S.LAL
Created: 30 Aug, 2010; Last Modified: 29 Apr, 2018
Algebra II - 03
Logarithms
A logarithm (abbreviated log) is an exponent to which a constant is raised to obtain a given number.
Suppose there are three real number a, x and N such that
a
x
=
N
, then x is the base-a logarithm of N. The logarithm of N to the base a is written as
log
a
N
.
a
x
=
N
⇔
x
=
log
a
N
.......(1)
Thus,
5
3
=
125
⇔
log
5
125
=
3
. In plain terms, the logarithm of 125 to the base 5 is 3.
While
a
x
=
N
is called the exponential form, its counterpart
log
a
N
is called the logarithmic form.
log of 1 to any base
We know that
a
0
=
1
holds true for all a.
The logarithmic form of this identity is
log
a
1=0
. Thus, the log of 1 to any base is always 0.
e.g.
log
5
1=
log
10
1=
log
2
1=0
log of any number to same base
We know that
a
1
=
a
.
The logarithmic form of this identity is
log
a
a
=
1
. Thus, the log of any number to the same base is always 1.
e.g.
log
5
5
=
log
10
10
=
log
2
2
=
1
Find (i) the logarithm of 1000 to the base 10 (ii)
1
9
to the base 3. [(Bansal, Mathematics Class IX, p. 114 Ex. 1)]
(i) Let
log
10
1000
=
x
log
10
1000
=
x
⇒
10
x
=
1000
⇒
10
x
=
10
3
⇒
x
=
3
∴
log
10
1000
=
3
(ii) Let
log
3
(
1
9
)
=
x
log
3
(
1
9
)
=
x
⇒
3
x
=
1
9
=
1
3
2
⇒
3
x
=
3
−
2
⇒
x
=
−
2
∴
log
3
(
1
9
)
=
−
2
Find x if (i)
log
4
(
x
+
3
)
=
2
(ii)
log
x
64
=
3
2
(iii)
log
2
(
x
2
−
4
)
=
5
. [(Bansal, Mathematics Class IX, p. 114 Ex. 2 (ii), (iv), (v)) ]
| (i) |
log
4
(
x
+
3
)
=
2
⇒
4
2
=
x
+
3
⇒
x
=
16
−
3
=
13
|
| (ii) |
log
x
64
=
3
2
⇒
x
3
2
=
64
⇒
x
3
=
64
⇒
x
3
=
(
8
2
)
2
=
8
4
=
(
2
3
)
4
=
(
2
4
)
3
⇒
x
=
2
4
=
16
|
| (iii) |
log
2
(
x
2
−
4
)
=
5
⇒
2
5
=
x
2
−
4
⇒
x
2
=
32
+
4
=
36
⇒
x
=
±
6
|
Laws of Logarithms
Product Law
The logarithm of a product is equal to the sum of logarithms of its factors.
log
a
(
m
×
n
×
...
)
=
log
a
m
+
log
a
n
+
...
.......(2)
Note:
log
a
(
m
+
n
)
≠
log
a
m
+
log
a
n
log
a
m
log
a
n
≠
log
a
m
−
log
a
n
Quotient Law
The logarithm of a fraction is equal to the difference between the logarithm of the numerator and the logarithm of the denominator.
log
a
(
m
n
)
=
log
a
m
−
log
a
n
.......(3)
Power Law
The logarithm of a power of a number is equal to the logarithm of the number multiplied by the power.
log
a
(
m
)
n
=
n
log
a
m
.......(4)
Common Logarithms
Base-10 logarithms are also known as common logarithms or common logs. In equations, common logarithms are commonly denoted by writing "log" without the subscript 10 rather than as "
log
10
".
Show that
3
log
2
+
log
5
=
log
40
. [(Aggarwal & Aggarwal, Mathematics IX, p. 61, Ex. 4(i))]
TPT (To prove that)
3
log
2
+
log
5
=
log
40
LHS
=
3
log
2
+
log
5
=
log
2
3
+
log
5
=
log
(
2
3
×
5
)
=
log
40
≡
RHS
Hence proved.
Express
log
10
108
5
in terms of
log
10
2
and
log
10
3
. [(Bansal, Mathematics Class IX, p. 116 Ex. 3)]
log
10
108
5
=
log
10
(
108
)
1
5
=
1
5
(
log
10
(
2
2
×
3
3
)
)
=
1
5
(
2
log
10
2
+
3
log
10
3
)
Given
log
10
x
=
a
and
log
10
y
=
b
, Write down (i)
10
a
−
1
in terms of x (ii)
10
2
b
in terms of y>. (iii) If
log
10
P
=
2
a
−
b
, express P in terms of x and y. [(Bansal, Mathematics Class IX, p. 119 Ex. 9)]
| (i) |
We have
10
a
−
1
≡
10
a
10
1
but
log
10
x
=
a
⇔
10
a
=
x
Thus,
10
a
−
1
=
10
a
10
1
=
x
10
|
| (ii) |
We have
10
2
b
≡
(
10
b
)
2
but,
log
10
y
=
b
⇔
10
b
=
y
Thus,
10
2b
=
(
10
b
)
2
=
y
2
|
| (iii) |
log
10
P
=
2
a
−
b
⇔
10
2
a
−
b
=
P
⇔
P
=
10
2
a
10
b
=
(
10
a
)
2
10
b
=
x
2
y
|
The Characteristic and Mantissa of a logarithm
From the equation
10
x
=
N
, it is clear that common logarithms will not, in general, be integral, nor will they always be positive. For example, the logarithm of a number might have a value like 1.2432, or −2.3461.
The integral part of a logarithm is called the characteristic, while the fractional part is called the mantissa.
Finding the Characteristic
The characteristic of the logarithm of any number to base 10 can be written down by inspection. Here's how:
- For any number greater than unity (
N > 1), the characteristic of its logarithm one less than the number of digits in the integral part of the number.
Thus, for N = 34.233, the characteristic of its logarithm (log34.233) is 1. For N = 4.132, the characteristic of its logarithm (log4.132) is 1.
- For any number less than unity (
N < 1), the characteristic of its logarithm is negative and one more than the number of zeroes immediately following the decimal point. Negative characteristics are generally written with the a bar on top, rather than the negative sign in the front.
Thus, for N = 0.4, the characteristic of its logarithm (log0.4) is −1. For N = 0.0023, the characteristic of its logarithm (log0.0023) is −3, written as
3
¯
.
Finding the Mantissa
The mantissa of a logarithm is found from the log tables. The entire sequence of digits for the number N, disregarding the decimal point, is used for finding the mantissa.
The decimal point is immaterial for finding the mantissa. This means that for the numbers having the same significant digits, like N = 475 and N = 0.0475, the mantissa for their logarithms (log475 and log0.0475) will be the same.
Table 1: A section of a four-figure logarithm table.
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
Mean Differences |
| 1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
| .. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
| 20 |
3010 |
3032 |
3054 |
3075 |
3096 |
3118 |
3139 |
3160 |
3181 |
3201 |
2 |
4 |
6 |
8 |
11 |
13 |
15 |
17 |
19 |
| 21 |
3222 |
3243 |
3263 |
3284 |
3304 |
3324 |
3345 |
3365 |
3385 |
3404 |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
| 22 |
3424 |
3444 |
3464 |
3483 |
3502 |
3522 |
3541 |
3560 |
3579 |
3598 |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
15 |
17 |
| .. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
.. |
Given in Table is a section of a four-figure logarithm table. Let us see how to find the mantissa from this table.
- The mantissa of
log2174:
Starting with the row headed by 21, locate the number in the column headed by 7. The number is 3365. To this number, add the number in the same row under the section “Mean Differences” and in the column headed by 4, which is 8. This gives 3365 + 8 = 3373. Since the mantissa is a fractional decimal, the mantissa is 0.3373.
- The mantissa of
log217:
Starting with the row headed by 21, locate the number in the column headed by 7. The number is 3365. Since all the digits have been accounted for, there is no need to venture into the section “Mean Differences”. The mantissa is, therefore, 0.3365.
- The mantissa of
log21:
Starting with the row headed by 21, locate the number in the column headed by 0. The number is 3222. The mantissa is, therefore, 0.3322.
- The mantissa of
log2:
Starting with the row headed by 20, locate the number in the column headed by 0. The number is 3010. The mantissa is, therefore, 0.3010.
- The mantissa of
log0.2:
Starting with the row headed by 20, locate the number in the column headed by 0. The number is 3010. The mantissa is, therefore, again, 0.3010.
- The mantissa of
log0.02:
Starting with the row headed by 20, locate the number in the column headed by 0. The number is 3010. The mantissa is, therefore, again, 0.3010.
The logarithm as a join of characteristic and mantissa
Now that we know how to obtain the characteristic and the mantissa of a logarithm, let us round up the discussion by finding the logarithm of the numbers mentioned.
log2174: characteristic is 3; mantissa is 0.3373.
Thus, the log is: 3 + 0.3373 = 3.3373.
log217: characteristic is 2; mantissa is 0.3365.
Thus, the log is: 2 + 0.3365 = 2.3365.
log21: characteristic is 1; mantissa is 0.3322.
Thus, the log is: 1 + 0.3322 = 1.3322.
log2: characteristic is 0; mantissa is 0.3010.
Thus, the log is: 0 + 0.3010 = 1.3010.
Now comes the interesting part:
log0.2: characteristic is -1, expressed as
1
¯
; mantissa is 0.3010.
Thus, the log is:
1
¯
+
0.3010
=
1
¯
.3010
Note that it is not −1 + 0.3010 = −0.6990.
log0.02: characteristic is -2, expressed as
2
¯
; mantissa is 0.3010.
Thus, the log is:
2
¯
+
0.3010
=
2
¯
.3010
Note that it is not −2 + 0.3010 = −1.6990.
The logarithm table system is built so as to have the advantage of having the mantissa always positive. A negative characteristic is denoted by a bar over it, indicating that in the characteristic and mantissa combination, only the characteristic can be negative. Thus,
2
¯
.3010
, the logarithm of 0.02, is very different from −2.3010, in which both the integral and the decimal part is negative.
There might be cases when the logarithm is completely negative. To convert it to the preferred format with mantissa as positive, add 1 to the decimal part and subtract 1 from the integral part.
Antilogarithm
Antilogarithm is the reverse of the logarithm, which essentially means that it is the original number itself from which the logarithm is derived. Antilogarithm is read from the antilog table in a way similar to reading the logarithm table.
To find the antilog of a logarithm, which consists of a characteristic and mantissa combination, the mantissa part is read off from the antilog-table to obtain a number. A decimal point is inserted into the number at a location dependent upon the characteristic of the logarithm.
So, when the characteristic is positive (say n), the decimal number is inserted after the (n + 1)th digit in the number. When it is negative
(
n
¯
), the number is preceded by n zeroes and a decimal point is placed before that.
As an example, to find the antilog of
2
¯
.2365
, a look at the antilog table for the mantissa .2365 gives the number 1724. Since the characteristic is
2
¯
, this means that a decimal point followed by two zeroes has to be prefixed to the number. So the antilog turns out to be 0.001724.
Use tables to evaluate
24.71
×
0.9246
512
correct to three decimal places. [ICSE 1994]
Let
x
=
24.71
×
0.9246
512
∴
log
x
=
log
(
24.71
×
0.9246
512
)
=
1
2
log
(
24.71
×
0.9246
512
)
=
1
2
(
log
24.71
+
log
0.9246
−
log
512
)
Looking up the log table, we have
log
x
=
1
2
(
1.3929
+
1
¯
.9660
−
2.7093
)
=
1
2
(
1.3929
+
(
−
1
+
0.9660
)
−
2.7093
)
=
1
2
(
−
1.3504
)
=
−
0.6752
=
(
−
1
+
1
)
−
0.6752
=
−
1
+
(
1
−
0.6752
)
=
−
1
+
0.3248
=
1
¯
.3248
Thus,
x
=
antilog
(
1
¯
.3248
)
Looking up the antilog table, we have,
x
=
antilog
(
1
¯
.3248
)
=
0.2113
The value correct to three decimal places is 0.211.
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List of References
Bansal, RK, Concise Mathematics I.C.S.E., Part I – Class IX, New Delhi: Selina Publishers, 2005.
Bibliography
Aggarwal, RS & Aggarwal, V, Secondary School Mathematics for Class IX, Patna, India: Bharati Bhawan, 1999.
Gibilisco, S, Everyday Math Demystified, USA: McGraw Hill, 2004.
Hall, HS & Knight, SR, Elementary Algebra for Schools, metric edn, Agra, India: AK Publications, 1966.
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