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LOGARITHMS © Department of Analytical Skills
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Content Introduction Properties of logarithms Problems
Problems using properties Application (Finding number of digits of a number) Solved problems Practice problems © Department of Analytical Skills
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1) Introduction What is log?
Subtraction is the inverse operation to addition. Ex: e + x = N ; x = N – e Division is the inverse operation to multiplication Ex: ex = N ; x = N/e Likewise Logarithm is the inverse operation to exponentiation. Ex: ex = N : x = loge N © Department of Analytical Skills
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1.i) Properties of Logarithms:
Note: If no base value is mentioned in log, then it is logarithms to the base 10 which are known as common logarithms. Property LHS RHS Addition property loga (xy) loga x + loga y Subtraction property loga (x/y) loga x - loga y Multiplication property loga (xn) n (loga x) Inverse property loga x 1 / logx a Application of multiplication and inverse loga^n (x) 1/n (loga x) Division property logb x / logb a Application of division property logx x 1 a^0 = 1 property loga 1 © Department of Analytical Skills
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2.i) Problems using properties
Example: Simplify log9√3243 log9√3243 = log√243243 = log243^1/2243 = 1/½ log243243 = 2 © Department of Analytical Skills
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Example 1. The value of log2 16 is: 1/8 4 8 16
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Example: If log 27 = 1.431, then the value of log 9 is: log 27 = 1.431
log 9 = log 3^2 = 2 log 3 = 2 (1.431/3) = 0.954 © Department of Analytical Skills
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Example 2. log √8 / log 8 is equal to: 1/8 1/4 1/2 8
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Example: Simplify log 135+3log133-3log133-3log1315+2log1365
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Example 3. 2log 7 – log 81 + log 189 – log 343 a)0 b) log 21 c) log 7
d) –log 3 © Department of Analytical Skills
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Example: If log10 7 = a, then log10 (1/70) is equal to:
log10 (1/70) = log log10 70 = log log log10 10 = 0 - a - 1 = - (a+1) © Department of Analytical Skills
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Example 4. If log10 2 = 0.3010, then log2 10 is equal to:
699/301 1000/301 0.3010 0.6990 © Department of Analytical Skills
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Example: find x, if (log 6561)/(log 243) = log 32 x+2
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Example 5. Solve the following for x: log 729 / log 81 = log x a)100
b) √1000 c) 1000 d) 10 © Department of Analytical Skills
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Example: If logx (9/16) = - 1/2 , then x is equal to:
- 3 4 B. 3 4 C D logx (9/16) = - 1/2 9/ = x^(-1/2) x = (9/16)^-2 = (16/9)^2 = 256/81 © Department of Analytical Skills
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Example 6. If logx16 = 0.8, then find x a)2 b) 4 c) 16 d) 32
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2.ii) Application (Finding the number of digits of a number)
Example: If log 3 = , the number of digits in 324 is: log 324 = 24 log 3 = 24 (0.4771) log 324 = 11.45 = = 12 digits (or) Number of digits in 324 = characteristics+ 1 = Characteristics is the part before decimal point © Department of Analytical Skills
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Example 7. If log 2 = 0.30103, the number of digits in 264 is:
18 19 20 21 © Department of Analytical Skills
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3) Solved examples © Department of Analytical skills
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1. Find the value of x: log3(log2(log2(log5x))) = 0 Solution:
= 625 © Department of Analytical Skills
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2. If log 2 = 0.3010 and log 3 = 0.4771, the value of log5 512 is:
Solution: log5 512 = log5 2^9 = 9 log5 2 = 9 log 2 / log 5 = 9 log 2 / log (10/2) = 9 log 2 / log 10 – log 2 = 9 log 2 / 1 – log 2 = / = /0.699 = 3.87 © Department of Analytical Skills
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4) Practice problems © Department of Analytical skills
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1. If ax = by, then: log a/b = x/y log a/log b = x/y
log a/ log b = y/x None of these © Department of Analytical Skills
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2. Simplify log3125525 + log125(25)60 a) 85 b) 145 c) 45 d) 145/4
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3. Simplify log √32(1/1024) a) 1 b) -1 c) -1/4 d) -4
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4. Simplify log 7 + 3 log 2 – log 14 – log 4 a) 0 b) log 14
c) 2 log 56 d) 1 © Department of Analytical Skills
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5. If log7(x-7) + log7(x^2+7x+49) =4 a)196 b) 7 c) 49 d) 14
© Department of Analytical Skills
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