Classwork Series and Exercises {Mathematics – SS1}: Logarithms

SSS 1 MATHEMATICS WEEKS 8 & 9

Topic: LOGARITHMS

From indices we have that 2³ = 8 where 2 is the base and 3 is the power. On the other hand, we can write that the log of 8 to base 2 is equal to 3, denoted thus

Log₂ 8 = 3

Also 5² = 25 which means that log₅25 = 2.

The log of any number N to base M is the index or power to which the base M must be raised to equal the number N.

i.e.    if x is the logarithm of a number N to base b then N = b^x

i.e.    if log_b N= x then N = b^x

e.g.    If       (i) log₂ 16 = 4, then 16 = 2⁴

(ii) log₅ 125 = 3, then 125 = 5³

(iii) log₉ 81 = 2, then 81 = 9²

(iv) log₂₅ 125 = 3/2 , then 125 = 25^3/2

(v) log₁₀ 1000 = 3, then 1000 = 10³

(iv) log₁₀ (1/10) = -1, then 1/10 = 10^-1

Conversely, if (i) 16 = 2⁴, then log₂ 16 = 4

(ii) 81 = 9², then log₉ 81 = 2

(iii) 125 = 25^3/2, then log₂₅ 125 = 3/2 and so on.

With the above expressions, we can say that logarithms and indices are inter-related.

Laws of Logarithms

These are similar to all the laws of indices.

It shows clearly that a logarithm is a mirror image of an index.

e.g.  100 = 10²; 2 = log₁₀100

1000 = 10³; 3 = log₁₀1000

0.01 = 10^-2; -2 = log₁₀^0.01

Evaluate the following:

(1) If

1. Product

Given that 100 = 100² log₁₀ 100 = 2 and 1000 = 10³  log₁₀ 1000 = 3 then

100 x 1000 = 10² x 10³ = 10²⁺³ = 10⁵ (1^st law, indices)

  log₁₀(100 x 1000) = 5 which is equal to 2+3

= log₁₀ 100 + log₁₀ 1000

Example

Given that log₁₀ 2 = 0.0310, log₁₀ 3 = 0.4771 and log₁₀ 7 = 0.8451, evaluate

Log₁₀ 42

Log₁₀ 42 = log₁₀ (7 x 6) = log₁₀ (7 x 2 x3)

= 0.8451 + 0.0310 + 0.4771 = 1.6232.

The logarithm of a product is the sum of the logarithms of the factors that make up the product.

2.Quotient

Since 1000 ¸ 100 = 10³ ¸ 10² = 10^3-2 = 10¹

 log₁₀(1000 ¸ 100) = 1 = log₁₀(1000) – log₁₀(100). This is the 2^nd law of indices.

Example

Log₁₀(14/3)

Log₁₀(14/3) = log₁₀ 14– log₁₀ 3

= log₁₀ (2 x 7) – log₁₀ 3

= log₁₀ 2 + log₁₀ 7 – log₁₀ 3

= (0.3010 + 0.8451) – 0.4771

= 0.6690

The logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.

3 Raising to a Power

Example

Log₁₀(100)² = 2log₁₀ 100 = 2 x 2

Evaluate log₁₀ 8, if log₁₀ 2 = 0.3010

log₁₀ 8   = log₁₀ 2³ = 3log₁₀ 2

= 3 x 0.3010

= 0.9030.

4. Roots

The logarithms of the n^th root of a number is the logarithm of the number, divided by n

e.g. ³ 1000 = ³ 10³ = ³ 10 x 10 x 10 = 10 or 10¹

 log ³ 1000 = 1 or (log 1000) ¸ 3 = 1

 log_b ⁿ x = 1/n log_b X.

Use of Log Tables

Express each number in the standard form. That is put a decimal after the first digit and multiply by the appropriate power of 10. The logarithm of a number has two parts the first of which is the exponent of 10, called the characteristic, and the second part is a decimal, called the mantissa, which is read from the log tables. The characteristic can be positive or negative. The negative characteristic is written with a bar above the number. The mantissa is always a positive decimal less than 1.

To multiply two numbers, find the logarithm of each number and add them. Then find the antilogarithm of the mantissa (the fraction part) from anti-log table (which will be a decimal between 1 and 10) and multiply by 10 raised to the characteristic to get the product.

The numbers	From the integral part, No. of digits -1	The std form of the Nos.	Characteristic part of the logarithm
389	3 – 1 = 2	3.99 x 102	2
458312	6 – 1 = 5	4.58312 x 105	5
17	2 – 1 = 1	1.7 x 101	1
1.4532	1 – 1 = 0	1.452 x 100	0

The mantissa part of the logarithm tables. Thus for the number 399 we look up for the first two digits 39 in the left hand column of the log table and follow the line across till we come under the third 9. Here we get 6010. Remember that the mantissa is always decimal so this is .6010. Now joining both the characteristics above with this mantissa part we have log 399 = 2.6010.

Similarly for 458312. Since our table is a four figure table only, this number becomes (to 4 significant figures) 458300. We look up 45 along the left hand column in the table and across under 8 we get 6609 and then the four digit 3 in the difference column. Across from 45 under 3 in the difference column is 3, we add this to 6609 to get 6612. Remember this is a decimal, i.e. .6612. Combining this with the characteristic 5 above we get that the logarithm of 458312 is 5.6612 and so on.

NB it is important to have logarithm table there with you for better understanding of the topic.

Antilogarithm

If the log₁₀ 1000 = then the antilogarithm of base 3 to 10 is 1000

Also if log₃ 81 = 4 then antilog₃ 1.3010 = 20.

Therefore the antilog of a number is that number whose logarithm is given ( i.e. the conversion of the logarithm). Also in common logarithms we usually drop the base 10 and just write log and not log₁₀.

Example

Find the number whose logarithm is 0.4771

Solution

We are looking for the antilog of the given log. From the antilog table we look for the position of the decimal parts in the antilogs, fro 0.4771 look for 47 under 7 in the main body of the antilog table we get 2999 then look under 1 in the difference column which gives 1, and 1 to 2999 to get 3000. Since the characteristic is 0 we add 1 to the characteristic when looking up in the antilog to get the number of digits before the decimal points in the antilog.

0 + 1 =0.

There is only one digit before the decimal point here. Putting the decimal point we have 3.000. The antilog of 0.4771 = 3.000 which is the required number

Multiply 256 by 768

256 = 2.56 x 10² and 768 = 2.56 x 10²

Log (256 x 768) = log 256 + log 768

= log (2.56 x 10²)+ log (2.56 x 10²) = 2.4082 + 2.8854 = 5.2963 =

256 x 768   = antilog 5.2963 = 1.966 x 10⁵ = 196600

Multiply 67846 and 0.0839

67846 = 6.7846 x 10⁴, 0.0839 = 8.39 x 10^-2

Log (67846 x 0.0839) = log 67846 +log 0.0839

= log (6.7846 x 10⁴)+ log (8.39 x 10^-2)

= 4.8315 + 2.9283 = 3. 7553

6.7846 x 0.0839 = antilog 3.7553

= 5.6925 x 10³ = 5692.5

The bar above the characteristic is written to show that only the characteristic part is negative and the mantissa part is positive.

Now let us do some problems on division.

4. Divide 826 by 347

826 = 8.26 x 10², 347 = 3.47 x 10²

Log (826 ¸ 347) = log (8.26 x 10²) – log (3.47 x 10²)

= 2.9170 – 2.5403

= 0.3767

826 ¸ 347     = antilog 0.3767

= 2.381 x 10⁰ = 2.381.

To divide a number by another number, find their logarithm and subtract the logarithm of the divisor from the logarithm of the dividend. Then find the antilogarithm of the mantissa from anti-log table and multiply by 10 raised to the characteristic to get the result.
5. Divide 273 by 9876

We can find square root of a number using log tables

6. Find the square root of 5468

5468        = 5468^1/2

Log 5468 = log 5864^1/2

                   = ½ log 5468 = ½ log (5.468 x 10³) = ½ 3.7378 = 1.8689

 5468     = antilog 1.8689 =7.394 x 10¹

=  73.94

To find the square root of a number, find its logarithm and divide it by 2 and then find its antilogarithm.

Hope you like this cool method of how to use log tables to multiply or divide two numbers or find the roots of numbers?

Questions:

1. Solve log₁₀ 6

A. 0.7778   B. 0.7781  C. 0.8778 D.  1.7778

2. Evaluate log₁₀(2/7)

A. 0.5441 B. 1.5442 C. 1.4327 D. 0.6551

Find from the tables of logarithms of

3. 1820

A. 7.2564 B. 2.6210 C. 3.2601 D. 3.5210

4. 236.9

A. 3.2674 B. 2.3749 C. 3.3749 D. 2.7359

5. Find the number whose logarithm is 1.3010

A. 21.00 B. 22.00 C. 23.00 D.20.00

Answers

1.  B

2.   A

3.   C

4.   B

5.   D