Logarithm - Examples – How to evaluate Logarithms
- Definition: The logarithm of a positive number x to the base b is the power to which b must be raised to obtain x.
- Written as: logb(x) = y if and only if by = x.
- Example: log2(8) = 3 since 23 = 8.
- Properties of Logarithms:
- logb(1) = 0 for any base b.
- logb(b) = 1 for any base b.
- logb(bx) = x for any base b and any positive number x.
- logb(xy) = logb(x) + logb(y) for any base b and positive numbers x and y.
- logb(x/y) = logb(x) - logb(y) for any base b and positive numbers x and y.
- logb(xn) = n * logb(x) for any base b, positive number x, and any real number n.
- Evaluation of Logarithms: Example 1: Evaluate log4(64)
- We need to find the power to which 4 must be raised to obtain 64.
- 43 = 64, so log4(64) = 3. Example 2: Evaluate log5(1/25)
- We need to find the power to which 5 must be raised to obtain 1/25.
- 5-2 = 1/25, so log5(1/25) = -2. Example 3: Evaluate log10(1000)
- We need to find the power to which 10 must be raised to obtain 1000.
- 103 = 1000, so log10(1000) = 3. Example 4: Evaluate log3(1)
- We need to find the power to which 3 must be raised to obtain 1.
- 30 = 1, so log3(1) = 0. Example 5: Evaluate log2(16)
- We need to find the power to which 2 must be raised to obtain 16.
- 24 = 16, so log2(16) = 4.
Logarithm - Examples – How to evaluate Logarithms
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- Example 6: Evaluate log6(36)
- We need to find the power to which 6 must be raised to obtain 36.
- 62 = 36, so log6(36) = 2.
- Example 7: Evaluate log7(49)
- We need to find the power to which 7 must be raised to obtain 49.
- 72 = 49, so log7(49) = 2.
- Example 8: Evaluate log8(64)
- We need to find the power to which 8 must be raised to obtain 64.
- 82 = 64, so log8(64) = 2.
- Example 9: Evaluate log9(81)
- We need to find the power to which 9 must be raised to obtain 81.
- 92 = 81, so log9(81) = 2.
- Example 10: Evaluate log10(100000)
- We need to find the power to which 10 must be raised to obtain 100000.
- 105 = 100000, so log10(100000) = 5.
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- Example 11: Evaluate log2(1/8)
- We need to find the power to which 2 must be raised to obtain 1/8.
- 2-3 = 1/8, so log2(1/8) = -3.
- Example 12: Evaluate log3(1/27)
- We need to find the power to which 3 must be raised to obtain 1/27.
- 3-3 = 1/27, so log3(1/27) = -3.
- Example 13: Evaluate log4(1/64)
- We need to find the power to which 4 must be raised to obtain 1/64.
- 4-3 = 1/64, so log4(1/64) = -3.
- Example 14: Evaluate log5(1/125)
- We need to find the power to which 5 must be raised to obtain 1/125.
- 5-3 = 1/125, so log5(1/125) = -3.
- Example 15: Evaluate log6(1/216)
- We need to find the power to which 6 must be raised to obtain 1/216.
- 6-3 = 1/216, so log6(1/216) = -3.
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- Example 16: Evaluate log2(√2)
- We need to find the power to which 2 must be raised to obtain √2.
- 21/2 = √2, so log2(√2) = 1/2.
- Example 17: Evaluate log3(√3)
- We need to find the power to which 3 must be raised to obtain √3.
- 31/2 = √3, so log3(√3) = 1/2.
- Example 18: Evaluate log4(√4)
- We need to find the power to which 4 must be raised to obtain √4.
- 41/2 = √4, so log4(√4) = 1/2.
- Example 19: Evaluate log5(√5)
- We need to find the power to which 5 must be raised to obtain √5.
- 51/2 = √5, so log5(√5) = 1/2.
- Example 20: Evaluate log6(√6)
- We need to find the power to which 6 must be raised to obtain √6.
- 61/2 = √6, so log6(√6) = 1/2.
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- Example 21: Evaluate log7(√49)
- We need to find the power to which 7 must be raised to obtain √49.
- 71/2 = √49, so log7(√49) = 1/2.
- Example 22: Evaluate log8(√64)
- We need to find the power to which 8 must be raised to obtain √64.
- 81/2 = √64, so log8(√64) = 1/2.
- Example 23: Evaluate log9(√81)
- We need to find the power to which 9 must be raised to obtain √81.
- 91/2 = √81, so log9(√81) = 1/2.
- Example 24: Evaluate log10(√100)
- We need to find the power to which 10 must be raised to obtain √100.
- 101/2 = √100, so log10(√100) = 1/2.
- Example 25: Evaluate log2(2-3)
- We need to find the power to which 2 must be raised to obtain 2-3.
- 2-3 = 1/8, so log2(2-3) = -3.
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- Example 26: Evaluate log3(32)
- We need to find the power to which 3 must be raised to obtain 32.
- 32 = 9, so log3(32) = 2.
- Example 27: Evaluate log4(41/2)
- We need to find the power to which 4 must be raised to obtain 41/2.
- 41/2 = 2, so log4(41/2) = 1/2.
- Example 28: Evaluate log5(52)
- We need to find the power to which 5 must be raised to obtain 52.
- 52 = 25, so log5(52) = 2.
- Example 29: Evaluate log6(62)
- We need to find the power to which 6 must be raised to obtain 62.
- 62 = 36, so log6(62) = 2.
- Example 30: Evaluate log10(10-2)
- We need to find the power to which 10 must be raised to obtain 10-2.
- 10-2 = 1/100, so log10(10-2) = -2.
Logarithm - Examples – How to evaluate Logarithms Definition : The logarithm of a positive number x to the base b is the power to which b must be raised to obtain x. Written as: log b (x) = y if and only if b y = x. Example: log 2 (8) = 3 since 2 3 = 8. Properties of Logarithms: log b (1) = 0 for any base b. log b (b) = 1 for any base b. log b (b x ) = x for any base b and any positive number x. log b (xy) = log b (x) + log b (y) for any base b and positive numbers x and y. log b (x/y) = log b (x) - log b (y) for any base b and positive numbers x and y. log b (x n ) = n * log b (x) for any base b, positive number x, and any real number n. Evaluation of Logarithms :
Example 1: Evaluate log 4 (64) We need to find the power to which 4 must be raised to obtain 64. 4 3 = 64, so log 4 (64) = 3.
Example 2: Evaluate log 5 (1/25) We need to find the power to which 5 must be raised to obtain 1/25. 5 -2 = 1/25, so log 5 (1/25) = -2.
Example 3: Evaluate log 10 (1000) We need to find the power to which 10 must be raised to obtain 1000. 10 3 = 1000, so log 10 (1000) = 3.
Example 4: Evaluate log 3 (1) We need to find the power to which 3 must be raised to obtain 1. 3 0 = 1, so log 3 (1) = 0.
Example 5: Evaluate log 2 (16) We need to find the power to which 2 must be raised to obtain 16. 2 4 = 16, so log 2 (16) = 4.