1. LOGARITHMS
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2. Content Introduction Properties of logarithms Problems
Problems using properties
Application (Finding number of digits of a number)
Solved problems
Practice problems
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3. 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
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4. 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
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5. 2.i) Problems using properties
Example: Simplify log9√3243
log9√3243 = log√243243
= log243^1/2243
= 1/½ log243243
= 2
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6. Example 1. The value of log2 16 is: 1/8 4 8 16
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7. 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
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8. Example 2. log √8 / log 8 is equal to: 1/8 1/4 1/2 8
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9. Example: Simplify log 135+3log133-3log133-3log1315+2log1365
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10. Example 3. 2log 7 – log 81 + log 189 – log 343 a)0 b) log 21 c) log 7
d) –log 3
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11. 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)
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12. Example 4. If log10 2 = 0.3010, then log2 10 is equal to:
699/301
1000/301
0.3010
0.6990
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13. Example: find x, if (log 6561)/(log 243) = log 32 x+2
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14. Example 5. Solve the following for x: log 729 / log 81 = log x a)100
b) √1000
c) 1000
d) 10
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15. 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
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16. Example 6. If logx16 = 0.8, then find x a)2 b) 4 c) 16 d) 32
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17. 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
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18. Example 7. If log 2 = 0.30103, the number of digits in 264 is:18 19 20 21
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19. 3) Solved examples
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20. 1. Find the value of x: log3(log2(log2(log5x))) = 0 Solution:
= 625
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21. 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
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22. 4) Practice problems
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23. 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
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24. 2. Simplify log3125525 + log125(25)60 a) 85 b) 145 c) 45 d) 145/4
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25. 3. Simplify log √32(1/1024) a) 1 b) -1 c) -1/4 d) -4
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26. 4. Simplify log 7 + 3 log 2 – log 14 – log 4 a) 0 b) log 14
c) 2 log 56
d) 1
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27. 5. If log7(x-7) + log7(x^2+7x+49) =4 a)196 b) 7 c) 49 d) 14
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