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Scientific Notation (Exercise 2.1)
Express in scientific notation:
(i) 2000000 → 2.0 × 10^6
(ii) 48900 → 4.89 × 10^4
(iii) 0.0042 → 4.2 × 10^{-3}
(iv) 0.0000009 → 9.0 × 10^{-7}
(v) 73 × 10^3 → 7.3 × 10^4
(vi) 0.65 × 10^2 → 6.5 × 10^1
Express in ordinary notation:
(i) 8.04 × 10^2 → 804
(ii) 3 × 10^5 → 300000
(iii) 1.5 × 10^{-2} → 0.015
(iv) 1.77 × 10^7 → 17700000
(v) 5.5 × 10^{-6} → 0.0000055
(vi) 4 × 10^{-5} → 0.00004
- Speed of light 3 × 10^8 m/s → 300000000 m/s
- Earth circumference 40075000 m → 4.0075 × 10^7 m
- Mars diameter 6.779 × 10^3 km → 6779 km
- Earth diameter 1.2756 × 10^4 km → 12756 km
Logarithmic Form (Exercise 2.2)
Express in logarithmic form:
(i) 10^3 = 1000 → log_{10} 1000 = 3
(ii) 2^8 = 256 → log_2 256 = 8
(iii) 3^{-3} = 1/27 → log_3 (1/27) = -3
(iv) 20^2 = 400 → log_{20} 400 = 2
(v) 16^{-1/4} = 1/2 → log_{16} (1/2) = -1/4
(vi) 11^2 = 121 → log_{11} 121 = 2
(vii) p = q^r → log_q p = r
(viii) 32^{-1/5} = 1/2 → log_{32} (1/2) = -1/5
Express in exponential form:
(i) log_5 125 = 3 → 5^3 = 125
(ii) log_2 16 = 4 → 2^4 = 16
(iii) log_{23} 1 = 0 → 23^0 = 1
(iv) log_5 5 = 1 → 5^1 = 5
(v) log_2 (1/8) = -3 → 2^{-3} = 1/8
(vi) log_9 3 = 1/2 → 9^{1/2} = 3
(vii) log 100000 = 5 → 10^5 = 100000
(viii) log_4 (1/16) = -2 → 4^{-2} = 1/16
Characteristics & Mantissa (Exercise 2.3)
Find characteristic:
(i) 5287 → 3
(ii) 59.28 → 1
(iii) 0.0567 → -2
(iv) 234.7 → 2
(v) 0.000049 → -5
(vi) 145000 → 5
Find logarithm:
(i) log 43 ≈ 1.6335
(ii) log 579 ≈ 2.7627
(iii) log 1.982 ≈ 0.2971
(iv) log 0.0876 ≈ -1.0575
(v) log 0.047 ≈ -1.3279
(vi) log 0.000354 ≈ -3.4510
From given log:
(i) log 3177 = 3.5019
(ii) log 31.77 = 1.5019
(iii) log 0.03177 = -1.4981
Find x (antilog):
(i) log x = 0.0065 → x ≈ 1.015
(ii) log x = 1.192 → x ≈ 15.55
(iii) log x = -3.434 → x ≈ 0.000368
(iv) log x = -1.5726 → x ≈ 0.02676
(v) log x = 4.3561 → x ≈ 22700
(vi) log x = -2.0184 → x ≈ 0.00959
Laws of Logarithms (Exercise 2.4)
Evaluate:
(i) log_2 18 – log_2 9 = 1
(ii) log_2 64 + log_2 2 = 7
(iii) (1/3) log_3 8 – log_3 18 = -2
(iv) 2 log 2 + log 25 = 2
(v) (1/3) log_4 64 + 2 log_5 25 = 5
(vi) log_3 12 + log_3 0.25 = 1
Single logarithm:
(i) (1/2) log 25 + 2 log 3 = log 45
(ii) log 9 – log (1/3) = log 27
(iii) log_5 b^2 * log_a 5^3 = 6 log_a b
(iv) 2 log_3 x + log_3 y = log_3 (x^2 y)
(v) 4 log_5 x – log_5 y + log_5 z = log_5 (x^4 z / y)
(vi) 2 ln a + 3 ln b – 4 ln c = ln (a^2 b^3 / c^4)
Expand:
(i) log (11/5) = log 11 – log 5
(ii) log_5 (8a^6)^{1/2} = (3/2) log_5 2 + 3 log_5 a
(iii) ln (a^2 b / c) = 2 ln a + ln b – ln c
(iv) log (xy / z)^{1/9} = (1/9) (log x + log y – log z)
(v) ln (16 x^3)^{1/3} = (4/3) ln 2 + ln x
(vi) log_2 (1 – a / b)^5 = 5 (log_2 (1 – a) – log_2 b)
Find x:
(i) log 2 + log x = 1 → x = 5
(ii) log_2 x + log_2 8 = 5 → x = 4
(iii) (81)^x = (243)^{x+2} → x = -10
(iv) ((1/27))^{x-6} = 27 → x = 5
(v) log (5x – 10) = 2 → x = 22
(vi) log_2 (x+1) – log_2 (x-4) = 2 → x = 17/3
Values with log table:
(i) (3.68 * 4.21) / 5.234 ≈ 2.954
(ii) 4.67 * 2.11 * 2.397 ≈ 23.458
(iii) ((20.46)^2 * 2.4122) / 754.3 ≈ 1.327
(iv) (9.364 * 21.64 / 3.21)^{1/3} ≈ 14.138
- Earthquake magnitude: 3
- Investment double: 14.2 years
- Temperature at 500m: 17.18°C
Review Exercise 2
MCQs:
(i) c) 5200000
(ii) b) 3.4 × 10^{-4}
(iii) b) 10
(iv) d) 3
(v) a) 2
(vi) c) 2.3010
(vii) d) undefined
(viii) c) 4
(ix) d) log 15
(x) c) log_3 81 = 4
Scientific notation:
(i) 5.67 × 10^{-4}
(ii) 7.34 × 10^2
Ordinary notation:
(i) 2600
(ii) 0.0008794
(iii) 0.000006
Logarithmic form:
(i) log_3 2187 = 7
(ii) log_a c = b
(iii) log_12 144 = 2
Exponential form:
(i) 4^x = 8
(ii) 9^3 = 729
(iii) 4^5 = 1024
Find x:
(i) x = 3
(ii) x = -1/2
(iii) x = -0.6
Single logarithm:
(i) log (x^7 / y^6)
(ii) log 2
(iii) log_5 2
Expand:
(i) log x + log y + 6 log z
(ii) (5/6) log_3 m + (1/2) log_3 n
(iii) (3/2) log 2 + (3/2) log x
Values:
(i) ≈4.085
(ii) ≈1133
(iii) ≈24.02
- Population: 2035
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Keywords
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