Study notes helpful for all, Exercises of Mathematics

Download Study notes helpful for all and more Exercises Mathematics in PDF only on Docsity!

CHAPTER

LOGARITHMS

Animation 3.1:Laws of logarithms Source & Credit: eLearn.punjab

version: 1.

Version: 1.1^ Version: 1.

Students Learning Outcomes

After studying this unit, the students will be able to:

• express a number in standard form of scientific notation and vice versa.
• define logarithm of a number y to the base a as the power to which a must be raised to give the number (i.e., a x^ = y ⇔ log ay = x , a > 0, - a ≠ 1 and y > 0).
• define a common logarithm, characteristic and mantissa of log of a number.
• use tables to find the log of a number.
• give concept of antilog and use tables to find the antilog of a number.
• differentiate between common and natural logarithm.
• prove the following laws of logarithm
  • log a ( mn ) = log am + log an ,
  • log a ( ) = log am – log an ,
  • log am n^ = n log am ,
  • log am log m n = log an.
• apply laws of logarithm to convert lengthy processes of multiplication, division and exponentiation into easier processes of addition and subtraction etc.

Introduction

The difficult and complicated calculations become easier by using logarithms. Abu Muhammad Musa Al Khwarizmi first gave the idea of logarithms. Later on, in the seventeenth century John Napier extended his work on logarithms and prepared tables for logarithms He used “ e ” as the base for the preparation of logarithm tables. Professor Henry Briggs had a special interest in the work of John Napier. He prepared logarithim tables with base 10. Antilogarithm table was prepared by Jobst Burgi in 1620 A.D.

3.1 Scientific Notation

There are so many numbers that we use in science and technical work that are either very small or very large. For instance, the distance from the Earth to the Sun is 150,000,000 km approximately and a hydrogen atom weighs 0.000,000,000,000,000,000,000,001,7 gram. While writing these numbers in ordinary notation (standard notation) there is always chance of making an error by omitting a zero or writing more than actual number of zeros. To overcome this problem, scientists have developed a concise, precise and convenient method to write very small or very large numbers, that is called scientific notation of expressing an ordinary number. A number written in the form a x 10 n , where 1 < a < 10 and n is an integer, is called the scientific notation. The above mentioned numbers (in 3.1) can be conveniently written in scientific notation as 1.5 x 108 km and 1.7 x 10 -24^ gm respectively.

Example 1 Write each of the following ordinary numbers in scientific notation (i) 30600 (ii) 0.

Solution 30600 = 3.06 x 104 (move decimal point four places to the left) 0.000058 = 5.8 x 10 -5^ (move decimal point five places to the right)

Observe that for expressing a number in scientific notation (i) Place the decimal point after the first non-zero digit of given number. (ii) We multiply the number obtained in step (i), by 10 n^ if we shifted the decimal point n places to the left (iii) We multiply the number obtained in step (i) by 10 -n^ if we shifted the decimal point n places to the right. (iv) On the other hand, if we want to change a number from scientific notation to ordinary (standard) notation, we simply reverse the process.

m n

Version: 1.1^ Version: 1.

3.2.2 Definitions of Common Logarithm, Characteristic

and Mantissa Definition of Common Logarithm

In numerical calculations, the base of logarithm is always taken as 10. These logarithms are called common logarithms or Briggesian logarithms in honour of Henry Briggs, an English mathematician and astronomer, who developed them.

Characteristic and Mantissa of Log of a Number Consider the following 103 = 1000 ⇔ log 1000 = 3 102 = 100 ⇔ log 100 = 2 101 = 10 ⇔ log 10 = 1 100 = 1 ⇔ log 1 = 0 10 –1^ = 0.1 ⇔ log 0.1 = – 10 –2^ = 0.01 ⇔ log 0.01 = – 10 –3^ = 0.001 ⇔ log 0.001 = –

Note:

Also consider the following table

Observe that

(i) Characteristic of Logarithm of a Number > 1 The first part of above table shows that if a number has one digit in the integral part, then the characteristic is zero; if its integral part has two digits, then the characteristic is one; with three digits in the integral part, the characteristic is two, and so on. In other words, the characteristic of the logarithm of a number greater than 1 is always one less than the number of digits in the integral part of the number. When a number b is written in the scientific notation, i.e., in the form b = a x10 n^ where 1 < a < 10, the power of 10 i.e., n will give the characteristic of log b.

Examples

By convention, if only the common logarithms are used throughout a discussion, the base 10 is not written.

For the numbers the logarithm is

Between 1 and 10 a decimal Between 10 and 100 1 + a decimal Between 100 and 1000 2 + a decimal

Between 0.1 and 1 –1 + a decimal Between 0.01 and 0.1 –2 + a decimal Between 0.001 and 0.01 –3 + a decimal

The logarithm of any number consists of two parts: (i) An integral part which is positive for a number greater than 1 and negative for a number less than 1, is called the characteristic of logarithm of the number. (ii) A decimal part which is always positive, is called the mantissa of the logarithm of the number.

Number Scientific Notation Characteristic of the Logarithm 1.02 1.02 x10^0 99.6 9.96 x 101 1 102 1.02 x 102 2 1662.4 1.6624 x 103 3

Version: 1.1^ Version: 1.

Characteristic of Logarithm of a Number < 1 The second part of the table indicates that, if a number has no zero immediately after the decimal point, the characteristic is –1; if it has one zero immediately after the decimal point, the characteristic is –2; if it has two zeros immediately after the decimal point, the characteristic is –3; etc. In other words, the characteristic of the logarithm of a number less than 1, is always negative and one more than the number of zeros immediately after the decimal point of the number.

Example Write the characteristic of the log of following numbers by expressing them in scientific notation and noting the power of 10. 0.872, 0.02, 0.

Solution

When a number is less than 1, the characteristic of its logarithm is written by convention, as 3, 2 or 1 instead of -3, -2 or -1 respectively (3 is read as bar 3 ) to avoid the mantissa becoming negative.

Note: 2.3748 does not mean - 2.3748. In 2.3748, 2 is negative but. is positive; Whereas in - 2.3748 both 2 and .3748 are negative.

(ii) Finding the Mantissa of the Logarithm of a Number While the characteristic of the logarithm of a number is written merely by inspection, the mantissa is found by making use of

logarithmic tables. These tables have been constructed to obtain the logarithms up to 7 decimal places. For all practical purposes, a four- figure logarithmic table will provide sufficient accuracy. A logarithmic table is divided into 3 parts. (a) The first part of the table is the extreme left column headed by blank square. This column contains numbers from 10 to 99 corresponding to the first two digits of the number whose logarithm is required. (b) The second part of the table consists of 10 columns, headed by 0, 1, 2, ...,9. These headings correspond to the third digit from the left of the number. The numbers under these columns record mantissa of the logarithms with decimal point omitted for simplicity. (c) The third part of the table further consists of small columns known as mean differences columns headed by 1, 2, 3, ...,9. These headings correspond to the fourth digit from the left of the number. The readings of these columns are added to the mantissa recorded in second part ( b ) above.

When the four-figure log table is used to find the mantissa of the logarithm of a number, the decimal point is ignored and the number is rounded to four significant figures.

3.2.3 Using Tables to find log of a Number

The method to find log of a number is explained in the following examples. In the first two examples, we shall confine to finding mantissa only.

Example 1 Find the mantissa of the logarithm of 43.

Solution Rounding off 43.254 we consider only the four significant digits 4325

Number Scientific Notation Characteristic of the Logarithm 0.872 8.72 x 10 -1^ - 1 0.02 2.0 x 10 -2^ - 2 0.00345 3.45 x 10 -3^ - 3

Version: 1.1^ Version: 1.

Example Find the numbers whose logarithms are (i) 1.3247 (ii) 2.

Solution (i) 1. Reading along the row corresponding to .32 (as mantissa = 0.3247), we get 2109 at the intersection of this row with the column corresponding to 4. The number at the intersection of this row and the mean difference column corresponding to 7 is 3. Adding 2109 and 3 we get 2112. Since the characteristic is 1 it is increased by 1 (because there should be two digits in the integral part) and therefore the decimal point is fixed after two digits from left in 2112. Hence antilog of 1.3247 is 21.12. (ii) 2. Proceeding as in (i) the significant figures corresponding to the mantissa 0.1324 are 1356. Since the characteristic is 2, its numerical value 2 is decreased by 1. Hence there will be one zero after the decimal point. Hence antilog of 2.1324 is 0.01356.

EXERCISE 3.

1. Find the common logarithm of each of the following numbers. (i) 232.92 (ii) 29. (iii) 0.00032 (iv) 0.
2. If log 31.09 = 1.4926, find values of the following (i) log 3.109, (ii) log 310.9, (iii) log 0.003109, (iv) log 0.3109 without using tables.
3. Find the numbers whose common logarithms are (i) 3.5621 (ii) 1.
4. What replacement for the unknown in each of following will make the statement true? (i) log 3 81 = L (ii) log a 6 = 0.

(iii) log 5 n = 2 (iv) 10 p^ = 40

5. Evaluate

6. Find the value of x from the following statements.

(i) log 2 x = 5 (ii) log 819 = x (iii) log 648 =

(iv) logx 64 = 2 (v) log 3 x = 4

3.3 Common Logarithm and Natural Logarithm

In 3.2.2 we have introduced common logarithm having base

10. Common logarithm is also known as decadic logarithms named

after its base 10. We usually take logx to mean log 10 x , and this type

of logarithm is more convenient to use in numerical calculations. John Napier prepared the logarithms tables to the base e. Napier’s logarithms are also called Natural Logarithms He released the first

ever log tables in 1614. log e x is conventionally given the notation In x.

In many theoretical investigations in science and engineering, it is often convenient to have a base e , an irrational number, whose value is 2.7182818...

3.4 Laws of Logarithm

In this section we shall prove the laws of logarithm and then apply them to find products, quotients, powers and roots of numbers.

x

Version: 1.1^ Version: 1.

(i) log a ( mn ) = log a m + log a n Proof

Let log am = x and log an = y

Writing in exponential form a x^ = m and a y^ = n. ∴ a x^ % a y^ = mn i.e., a x +y^ = mn

or log a ( mn ) = x + y = log am + log an

Hence log a ( mn ) = log am + log an

Note: (i) log a ( mn ) ≠ log am x log an (ii) log am + log an ≠ log a ( m + n) (iii) log a ( mnp ...) =log am + log an + log ap + … The rule given above is useful in finding the product of two or more numbers using logarithms. We illustrate this with the following examples.

Example 1 Evaluate 291.3 % 42.

Solution

Let x = 291.3 % 42.

Then log x = log (291.3 % 42.36)

= log 291.3 + log 42.36 , (log amn = log am + log an ) = 2.4643 + 1.6269 = 4.

x = antilog 4.0912 = 12340

Example 2 Evaluate 0.2913 % 0.004236.

Solution Let y = 0.2913 % 0. Then log y = log 0.2913 + log 0. = 1.4643 + 3. = 3.

Hence y = antilog 3.0912 =0.

Proof

Let log am = x and log an = y

Then a x^ = m and a y^ = n

Note:

Example 1

Solution

Thus x = antilog 0.8374 = 6.

Note that log a a = 1

Version: 1.1^ Version: 1.

Note:

EXERCISE 3.

1. Write the following into sum or difference

2. Express log x – 2 log x + 3 log (x + 1) – log (x^2 – 1) as a single

logarithm.

3. Write the following in the form of a single logarithm. (i) log 21 + log 5 (ii) log 25 – 2 log 3

(iii) 2 log x – 3 log y (iv)^ log^5 +^ log^6 –^ log

4. Calculate the following: (i) log 32 % 1og 281 (ii) log 53 % 1og 325
5. If log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, then find the values of the following (i) log 32 (ii) log 24 (iii) log (iv) log (v) log 30

3.5 Application of Laws of Logarithm in Numerical Calculations

So far we have applied laws of logarithm to simple type of products, quotients, powers or roots of numbers. We now extend their application to more difficult examples to verify their effectiveness in simplification.

Example 1 Show that

Solution

= 7[log 16 - log 15] + 5[log 25 - log 24] + 3[log 81 - log 80] = 7[log 24 - log (3 x 5)] + 5[log 52 - log (2^3 x 3)] + 3[log 34 - log (2^4 x 5)] = 7[4 log 2 - log 3 - log 5] + 5[2 log 5 - 3 log 2 - log 3] + 3[ log 3 - 4 log2 - log 5] = (28 - 15 - 12) log 2 + (- 7 - 5 + 12) log 3 + (- 7 + 10 - 3)log 5 = log 2 + 0 + 0 = log 2 = R.H.S. Example 2 Evaluate:

Solution

(i) During conversion the product form of the change of base rule may often be convenient. (ii) Logarithms can be defined to any positive base other than 1, e or 10, and are useful for solving equations in which the unknown appears as the exponent of some other quantity.

Version: 1.1^ Version: 1.

or y = antilog 1.4762 = 0.

Example 3 Given A = A (^) o e - kd. If k = 2, what should be the value of d to make

Solution Given that A = A (^) o e - kd. ⇒

Substituting k = 2, and , we get = e-^2 d Taking common log on both sides, log 101 - log 102 = - 2 d log 10 e , where e = 2. 0 - 0.3010 = - 2 d (0.4343)

EXERCISE 3.

1. Use log tables to find the value of
   2. A gas is expanding according to the law pvn^ = C. Find C when p = 80, v = 3.1 and
   3. The formula p = 90 (5)- q /10^ applies to the demand of a product, where q is the number of units and p is the price of one unit. How many units will be demanded if the price is Rs 18.00?
   4. If A = p r^2 , find A, when p = and r = 15
   5. If V = p r^2 h, find V, when p = , r = 2.5 and h = 4.

REVIEW EXERCISE 3

1. Multiple Choice Questions. Choose the correct answer.
2. Complete the following: (i) For common logarithm, the base is …….. (ii) The integral part of the common logarithm of a number is called the ….. (iii) The decimal part of the common logarithm of a number is called the ….

(iv) If x = log y , then y is called the ........... of x.

(v) If the charactcristic of the logarithm of a number is 2, that number will have ......... zero(s) immediately after the decimal point. (vi) If the characteristic of the logarithm of a number is 1, that number will have digits in its integral part.

3. Find the value of x in the following:

(i) log 3 x = 5 (ii) log 4256 = x

(iii) (iv)

22 7 1 3

22 7

1 2