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1 Logarithm The purpose of this unit is to equip the learners with the concept of logarithm. Under the logarithm, the topics covered are nature of logarithm, laws of logarithm, change the base of logarithm, antilogarithm and its operation followed by ample examples.

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3 Bangladesh Open University Lesson-: Nature and Basic Laws of Logarithm After studying this lesson, you should be able to: Discuss the nature of logarithm; Identify the basic laws of operation of logarithm; Explain the characteristics and mantissa of logarithm. Meaning of a Logarithm Logarithm is the important tool of modern mathematics. If a x n, then x is said to be the logarithm of the number 'n' to the base 'a'. Symbolically it can be expressed as follows: log a n x. In this case a x n is an exponential form and loga n x is a logarithmic form. The object of logarithm is to make common calculations less laborious and the method consists in replacing multiplication by addition and division by subtraction. Logarithm to the base 'e' is called natural logarithm and when the base is 0, the logarithm is called common logarithm. For example, (i) 5 5 log 5 5, i.e. the logarithm of 5 to the base 5 is equal to. Logarithm to the base 'e' is called natural logarithm and when the base is 0, the logarithm is called common logarithm. (ii) ( 6 )6 log 6 6 equal to 6., i.e. the logarithm of 6 to the base is Similarly, Exponential form Logarithmic form 8 log log log 0 log 0 or Logarithmic form Exponential form log 6 6 log p R Q P Q R log log Business Mathematics Page-

4 School of Business Fundamental Properties and Laws of Logarithms The fundamental properties and laws of logarithm are as follows: () The logarithm of the production of two factors is equal to the sum of their logarithms; i.e., log a mn log a m + log a n. () The logarithm of quotient is equal to logarithm of the numerator m minus the logarithm of the denominator; i.e., log a loga m n log a n. () The logarithm of any power of a number is equal to the product of the index of the power and the logarithm of the number; i.e. log a m x x log a m. () Base changing formula: The formula which tells us how to change log a n from one base to another is : log b n log b i.e., (log b n) (log a b) log a n a The logarithm of a number consists of two parts. The first part is called characteristics and the second part are termed as mantissa. Characteristics and Mantissa of a Logarithm The logarithm of a number consists of two parts: (i) an integer positive, negative or zero (ii) a positive or negative proper fraction. The first part is called characteristics and the second part are termed as mantissa. Since 0 0 log log log log ,000 log 0,000 Similarly, since , log , log , log , log In general, the logarithm of a number containing n digits only in its integral part is {(n ) + a} fraction and the logarithm of a number having N zeros just after the decimal point is { (n+) + a} fraction. Let us take some examples on logarithm. Example-: Unit- Page-

5 Bangladesh Open University If log x 65 ; find the value of x. log x 65 can be expressed in exponential form as Example-: If x 65 or, x 5 or, x 5 5 log x 7 Expressing log x 7 Example-:, find the value of x. or, x or, x in the exponential form, we get ( 7 ) x or, x or, x 9 If 0 x 8, find the value of x. Here 0 x 8 can be expressed in logarithmic form as, log 0 8 x Therefore, x log (by using scientific calculator). Example-: The logarithm of a number is.5. Find the characteristics and mantissa. Let log N -.5 Example-5: ( 0.5) ( + 0.5) The characteristics is and mantissa is Find the logarithm whose logarithm is.678. Business Mathematics Page-

6 School of Business From the Anti-log Table, For mantissa 0.67, the number 9 For mean difference 8, the number 5 For mantissa 0.678, the number (9 + 5) 96. The characteristics is, therefore the number must have digits in the integral part. Hence, antilog Example-6: Find the number whose logarithm is.678. Let log N From Antilog Table, For mantissa 0.5, the number 0. For mean difference, the number For mantissa 0.5, the number (0 + ) 06 The characteristic is, therefore the number is less than one and there must be two zeros just after the decimal point. Hence, antilog Example-7: Find the value of (i) log 6; (ii) log 9 ; (iii) log9 (iv) log (i) Let log 6 x (ii) Let log 9 x or, 6 x or, 6 x x 6 or, 9 x or, 9 - x or, - x x (iii) Let log 9 x or 9 x or x or x (iv) Let log x or, x or, x or, - x Unit- Page-

7 Bangladesh Open University Example-8: x or, - x or, x x Find the logarithm of the following to the base indicated in brackets. (i) 7, (); (ii) 6, (8); (iii) 000, (0); (iv) 0.5, (). (i) 7 (ii) 6 8 log 7. log 8 6. Example-9: (iii) (iv) log log 0.5. Without using tables, evaluate log log 0 00 log 0 0 log log 0 70 log 0 + log 0 5 log 0 0 Example-0: Simplify 7 log log 0 9 log 5 5 log 5 + log log [log 0 log 9] [log 5 log ] + [log 8 log 80] 7[(log 5 + log ) log ] [log 5 (log + log )] + [log (log5 + log )] 7 log log log log 5 + log + 6 log + log log 5 log (7 ) log 5 + ( + ) log + (7 + 6 ) log log. Business Mathematics Page-5

8 School of Business Example-: Find the value of log 75 6 log 5 9 each logarithm. + log, when 0 is the base of log 75 6 log log [log 0 75 log 0 6] [log 0 5 log 0 9] + [log 0 log 0 ] [(log log 0 ) log 0 ] [log 0 5 log 0 ]+ [(log 0 + log 0 ) log 0 5 ] log log 0 log 0 log log 0 + log 0 + log 0 5 log 0. log 0 Example-: Prove that, (log ).(log ).(log 5 ).(log 6 5 ).(log 7 6 ).(log 8 7 ) log log log 5 log 6 log 7 log8 L.H.S. log log log log 5 log 6 log 7 log 8 log log log log log Therefore, L.H.S R.H.S. (Proved). Example-: Solve the equation: log 0 (x+) log 0 x log 0 (5x ) log 0 (x+) log 0 x log 0 (5x ) or, log 0 (x+) + log 0 (5x ) log 0 x (x + )(5x ) or, log 0 x Unit- Page-6

9 Bangladesh Open University (x + )(5x ) 0 x or, 5x 9x + 0x 6 0x or, 5x + x 6 0 or, 5x + x 6 0 or, (5x+6) (x ) 0 x or 6 /5 since x cannot be negative, x. Example-: Show that x log y log z. y log z log x. z log x log y Let the left side N, then multiply both sides by log; we have log N log (x log y log z ) + log (y log z log x ) + log (z log x log y ) (log y log z) log x + (log z log x) log y + (log x log y) log z log y. log x log z. log x + log z. log y log x. log y + log x. log z log y. log z 0 Therefore N 0 0 Hence L. H. S R. H. S. (Proved). Example-5: Find the value of log 7 9; if log and log Here log 7 9 can be written (by changing base) as log log log 7 log log log (0.85).690 (0.77)..8 Example-6: Prove that log 7 6 log. log 8 9 Business Mathematics Page-7

10 School of Business By changing all logarithms on LHS to the base 0 by using the formula, we get Here log log 7 log 7 log log log 8 log 8 log log 9 log 7 log.log 8 9 log log 9 log 5 log7 log log7 5 5 log log 7 5 log log 7 log 7 (log 8 log 9 ) 5 log log7 log log log 7 6 (Proved) 5 log Unit- Page-8

11 Bangladesh Open University Questions for Review These questions are designed to help you assess how far you have understood and can apply the learning you have accomplished by answering (in written form) the following questions:. Define logarithm. Is there any distinction between natural and common logarithm?. What are the fundamental rules of logarithmic operations?. Find the value of log log 0 0 log0 6. If log and log ; find (i) log 0 5; and (ii) log If log x 8 ; find the value of x. 6. Evaluate log + log 9 log 6 + log 7 log If log a 0.589; log b.856 and log c.96; find the value of a b log c 8. Find the value of log 0 5 log 0 + log If log 0.77; log 0.00 and log , find the value of log If log 0 [98+ x x + 6 ], find the value of x.. Show that log + 6 log log 5. Solve log 0 (7x 9) + log 0 (x ). + 7 log Prove that log 0 9 log log 8 80 log Multiple Choice Questions ( the appropriate answer). If a x b, then (a) log b x a (b) log a x b (c) log a b x. If log a b c; then (a) b c a (b) a c b (c) a b c. Business Mathematics Page-9

12 School of Business. The value of log 5 65 is (a) (b) (c). The value of log 6 is (a) (b) 8 (c) If log 8 x, then the value of x is (a) (b) (c). 6. The value of [log 5 + log log ] is equal to: (a) log 5 (b) log (c) If log 0.00 and 5 x 00; then x is equal to: (a).0 (b).7 (c).6 8. The value of [log a bc + log b ac + log c ab ] is (a) 0 (b) (c) abc 9. If log 0 x, the value of x is (a) 5 (b) 00 (c) 5 0. The characteristic in log ( ) is (a) 5 (b) (c). The Mantissa of log 7 is.550. The value of log 0.7 is (a) (b).550 (c).550 Unit- Page-0

13 Bangladesh Open University Lesson-: Natural Logarithm and Antilogarithm After studying this lesson, you should be able to Explain the natural logarithm; Explain antilogarithm;. Apply the principles of logarithm to solve the mathematical problems. Nature of Natural Logarithm rithms to the 'e' are known as ral logarithms. value of 'e' may lculated from the ries. Logarithms to the base 'e' are known as natural logarithms. The value of 'e' may be calculated from the 'e' series, where e +! +! +!... [Here! is factorial, where n! n(n-) (n-)... 0!] Hence! 0! (since 0! ) Again, 6! 6 5 0! 6 5! From 'e' series, the value of 'e' is.788. Let, e x N or, log e N x When the base of logarithm is 'e', it may be expressed is ln; i.e., log e N ln N. Again, log 0 N log en log e 0 (through change of base) or, log N InN In0 In N log N ln 0 Again, In 0 log 0 log e log e In N log N log e or, log N In N log e g scientific ulator we can y find the value of ased number. Using scientific calculator we can easily find the value of 'e' based number: For example, log e log e log e 0.05 Business Mathematics Page-

14 School of Business log e e In e. Let us take same examples. Example-: Find the value of n, if (.08) n. Given, (.08) n or, In(.08) n In or, n In (.08) In ln.0986 n ln (.08) Example-: Find the value of i, if (+i) Here (+i) or, In (+i) In or, In (+i) In.7 (App.) ln 0.69 or, In (+i) or, (+i) e or, i i Anti-logarithm Let log a N x, then N is called the anti-logarithm of x to the base a and is written in short as antilog a x. If log a N x, then N antilog a x For example, if log 000, then antilog 000 If log , then antilog Let log a N x, then is called the antilogarithm of x to the base a and is written in short as antilog x Example-: Find the number whose logarithm is.78 Unit- Page-

15 Bangladesh Open University Let the number is x Therefore, log x.78 or, x antilog.78 x 5.90 (by using calculator). Example-: Find the value of ( ) Let x log x log ( ) log log or, log x.76 x Antilog.76 6, Example-5: Solve the equation x. 7 x+ x+5 Taking logarithm of both sides, we have x log + (x+) log 7 (x+5) log or, x log + x log 7 + log 7 x log + 5 log or, x log + x log 7 x log 5 log log 7 or, x (log + log7 log ) 5 log log7 x 5log log 7 log + log 7 log (App.) Example-6: Evaluate by using logarithm Let x Taking logarithm of both sides, we have log x log log log (.5).607 Business Mathematics Page-

16 School of Business x antilog (.96) Example-7: Evaluate by using logarithm: ( ) 6.8. ( 6 ) (i) (ii) ( ) 6.8. ( 6 ) (i) Let x Taking logarithm of both sides, we have log x log (6.8) + log (6) log (0.005) or, log x [0.798] + [.795] (-.00) or, log x or, log x x antilog (.69) (ii) Let x Taking logarithm of both sides, we have log x log or, log x 7 or, log x 7 or, log x 7 or, log x 7 [log log log 6.] [ ] [ ] [.7] 0.0 x antilog ( 0.0) 0.66 So, x Example-8: Unit- Page-

17 Bangladesh Open University ( ) 5. ( ) Find the value of ( 80) ( ) 5. ( ) Let x ( 80) Taking logarithm of both sides, we have log x log 5 + log log 80 or, log x (.58).5798 or, log x or, log x.096 Hence, x antilog (.096) Example-9: Find the 7 th root of Let x Taking logarithm of both sides, we have log x 7 log ( ) 7 or, log x (.856) or, log x 0.69 or, x antilog ( 0.69) 0.0 (App.) Example-0: Find the value using logarithm, Let x.786 Taking logarithm of both sides we get log x log log 9.56 log.786 or, log x ( ) or, log x.909 Business Mathematics Page-5

18 School of Business x antilog Example-: Solve x. x 00 x. x 00 or, log ( x. x ) log 00 or, x log + x log log 0 or, x(0.00) + x(0.77) log 0 or, 0.00x x or,.55x x.557 Hence, x.598. Example-: Solve x x+ + 0 x x+ + 0 or, ( x ) x. + 0 Let x y y y + 0 or, y y y or, y (y ) (y ) 0 or, (y ) (y ) 0 Now either, y 0 or, y 0 or, y or, y or, x or, x or, log x log or, log x log or, x log log or, x log log or, x log log or, x log or, x or, x or, x 0 Unit- Page-6

19 Hence x or 0. Example-: Prove that, 7 log log log 5 5 L.H.S. 7 log + 6 log + 5 log log 5 Bangladesh Open University + log 5 log 7(log + log 5 log ) + 6( log log ) + 5(log log 5) + (5 log log 5) 7 log + 7 log 5 8 log + 8 log 6 log + 5 log 5 log log log 5 log (Proved) Business Mathematics Page-7

20 School of Business Questions for Review These questions are designed to help you assess how far you have understood and can apply the learning you have accomplished by answering (in written form) the following questions:. Find the value of (.96) 6.. Find the value of Find the value of log 6 + log / log 0.. Find the value of log / + log 5/ log Find the value of x; if log x + log x Evaluate by using logarithm Solve 0 x 5. x 5 x.7 x 8. Solve for x, if log x (8x ) log x. 9. Evaluate Multiple Choice Questions ( the appropriate answer). (log log 0 ) is equal to: (a) (b) 000 (c) If log (x+) + log (x ) log, then x is equal to (a) 7 (b) (c) 8. If log 0 5 +og 0 8 x; then x is equal to (a) (b) (c) /. The value of x satisfying log x 0.80 is: (a) 5.6 (b) 0 (c) If loga b-c logb c-a logc a-b, then the value of aa b b c c is (a) abc (b) abc (c). Unit- Page-8