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# Logarithms

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Logarithm lesson
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# Logarithms

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.

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.

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### Logarithms

1. 1. LOGARITHMS BirinderSingh,AssistantProfessor,PCTE
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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