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Advanced Mathematics Training Class Notes Chapter 7: Logarithm and Number System Chapter 7 Properties of Logarithm Logarithm Recall that: 1. am+n = am an and Number System 2. 3. 4. 5. Logarithm (對數 對數) 對數 Logarithm was originally invented for an easier multiplication and division. The use of logarithm is to change multiplication and division to addition and subtraction. How to do so? 2 21 4 22 8 23 16 24 32 25 64 26 128 27 am an a = 1 (for a ≠ 0) a1 = a anc = (ac)n 0 By these, we can immediately see that: 1. loga m + loga n = loga (m n) Take an example. It shouldn’t be hard to know 8 × 16 = 128. But doesn’t it also mean 23 × 24 = 27, or simply, “3 + 4 = 7”? Here we changed 8 into 3, 16 into 4. We calculated 3 + 4 = 7, and change 7 back to 128. In the process of changing 8 into 3, etc, we are taking logarithm. 1 20 am−n = 256 28 2. log a m − log a n = log a 3. 4. 5. loga 1 = 0 loga a = 1 loga cn = n loga c ( ) m n (Addition property) (Subtraction property) (Logarithm of 1) (Logarithm of base) (Index property) Moreover, y In general, if N can be written as x , then y is called the logarithm of N base x. It is written as y = logx N. For example, 3 = log2 8; -1 = log10 0.1. If the base x is 10, we call this as common logarithm (常用對數), and denoted just by “log”. For example, log 100 = 2. 6. log a b = log c b log c a (Base property) 7. log a b = 1 logb a (Inverse base property) 8. log a b = log a n b n 9. If the base is 2, it is sometimes denoted as “lg”. For example, lg 64 = 6. c log a a = a b log a c (Base index property) =c (Cancellation property) b log n a 10. a = n (Index to base property) These form the logarithm properties, which is useful in solving equations involving logarithms and indices. Besides these properties, you should also memorize the following values: log 2 = 0.301, log 3 = 0.477 Finally, if a number < 0 is taken logarithm (logn x, x < 0), the result is undefined. 41 Advanced Mathematics Training Class Notes Chapter 7: Logarithm and Number System Base-n Number System Converting Number between Binary and Decimal In daily life, we express numbers in decimal system (十進制), but sometimes it would be more useful in using other bases, like binary system (二進制) in computer science. Converting binary number to decimal number It is easy. Take an example, to convert 10111010012 to decimal: Label the numbers with 0, 1, 2, etc. from right hand side at the unit (個位): 9876543210 1011101001 A base-n number system is that, when you count numbers from 1, 2, 3, etc., you need to add one more digit when reaching n, and reset the current digit to 0. For example, in decimal (base-10): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, … Remove the column having “0” as the digit: 976530 111111 In binary (base-2): 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, … In ternary (base-3): 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120, 121, 122, 200, … Remove the last row: 9,7,6,5,3,0 Raise power of 2 from them: 512,128,64,32,8,1 Add them altogether: 745. This is what 10111010012 represents. If the base is higher than 10, like 16, the digits (0~9) are not enough. We will use alphabets (A~Z) in this case. For example, in hexadecimal (base-16): Another example: 1100.101012. 3 2 1 0 -1 -2 -3 -4 -5 11001 0 1 0 1 1, 2, 3, …, 8, 9, A, B, C, D, E, F, 10, 11, …, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20, … Base 2 3 4 5 8 10 16 3 2 -1 -3 -5 111 1 1 Name Binary (or “Bin”) Ternary Quaternary Quinary Octal (or “Oct”) Decimal (or “Dec”) Hexadecimal (or “Hex”) 23 + 22 + 2-1 + 2-3 + 2-5 = 12.6562510 Converting decimal number to binary number We use short division to do this. For example, change 12310 to binary. Divide 123 by 2, write down the remainder, and repeat until zero is reached: A base-n number is usually written end with “n”, like 1010112. A base-n number ( a j …a2a1a0 )n can be considered as a j n j + a j −1n j −1 + … + a2n 2 + a1n + a0 . 42 Advanced Mathematics Training Class Notes Chapter 7: Logarithm and Number System 2 123 …1 2 61 …1 2 30 … 0 2 15 …1 2 7 …1 2 3 …1 2 1 …1 0 Read from bottom to top, 1111011. This is the representation of 12310 in binary. Revision For decimal numbers with fractional part (小數部), we separate the fractional part from its integral part (整數部). Like 7.687510, the integral part is 7, and the fractional part is 0.6875. Convert the integral part into binary directly (1112), and do the fractional part as follows: Exercise 0.6875 ×2 = 1.375 …1 0.375 ×2 = 0.75 … 0 0.75 ×2 = 1.5 …1 ×2 = 1 0.5 …1 0 We multiply 2 to the original fractional part. If the resulting integral part is 1, take away that, and repeat the process with the remaining fractional part until reaching 0. Read from top, 1011. This is the fractional part in binary. On the whole, 7.687510 = 111.10112. 1. Given log 2 = a, log 3 = b. Express the followings in terms of a and b: a) log 5 b) log 9 c) log 270 d) lg 30 e) log5 6 2. Solve the following equations: a) log(2x – 4) = 3 b) log(x – 1) + log(x + 3) = log 5 c) log x – log (3x2 – 5) = -1 d) (log x)2 – log x2 = 3 e) 52x – 6(5x+1) + 125 = 0 3. Which is bigger: 275, 350, 525? Give reasons. 4. (HKMO 2004 Heat) Find log1400 2 + log1400 3 5 + log1400 6 7 . In this chapter, we’ve learnt: 1. What is logarithm 2. Properties of logarithm 3. Base-n number system 4. Converting number between binary and decimal 5. Application of base-n numbers in solving equations involving indices In the followings, if not specified, x is the variable, and k is a positive number. For the other bases, the converting process is similar, and these are left to the readers to examine. Application of Base-n Numbers in Solving Equations Involving Indices 1 1 + . a b w 6. (ISMC 2000 Final) Find integral solutions of 2 + 2x + 2y + 2z = 20.625, where w>x>y>z. 7. (ISMC 2000 Final) Find a sequence a that is composed of 14 positive integers such that: 5. (HKMO 2001 Heat) If 4a = 25b = 10, find out Base-n numbers other useful in solving equations like na + nb + nc + … + nz = X. It would be very easy if you convert X into base-n, which is nearly impossible using logarithm only. w x y 14 ∑ 3a k k =1 z For example, solve 2 + 2 + 2 + 2 = 42.5. Convert 42.5 into binary (101010.12), it is then clear that w = 5, x = 3, y = 1, z = -1. 43 = 6558 Advanced Mathematics Training Class Notes Chapter 7: Logarithm and Number System 8. (ISMC 2000 Final) Given: ( x = x + log ( x x2 = x1 + log x12 + x12 + 1 3 2 2 2 ) + x22 + 1 ) ⋮ ( x1 = x2000 + log x20002 + x20002 + 1 ) Find x2000. [Hint: Property #3] 44 Advanced Mathematics Training Class Notes Chapter 7: Logarithm and Number System Suggested Solutions for the Exercise 1a) 1 – a b) 2b c) 3b + 1 d) 1+ab e) b+a 1− a 2a) 502 b) –4, 2 5 ± 2 10 c) 3 d) 1/10, 1000 e) 1, 2 3) 350 4) 1/6 5) 2 6) 4, 2, -1, -3 7) 1,1,2,2,3,3,4,4,5,5,6,6,7,7 8) 0 45