This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Hapa kuna jinsi ya kutatua swali hili kwa kutumia sheria za logariti na thamani ulizopewa: Tunapewa: _10 2 = 0.3010 _10 3 = 0.4771 a) Tathmini _10 18 Step 1: Andika 18 kama bidhaa ya nambari kuu (prime factors). 18 = 2 × 9 = 2 × 3^2 Step 2: Tumia sheria ya logariti ya kuzidisha (_b (MN) = _b M + _b N) na sheria ya nguvu (_b (M^p) = p _b M). _10 18 = _10 (2 × 3^2) = _10 2 + _10 3^2 = _10 2 + 2 _10 3 Step 3: Badilisha thamani ulizopewa. _10 18 = 0.3010 + 2 × 0.4771 Step 4: Fanya hesabu. _10 18 = 0.3010 + 0.9542 = 1.2552 Jibu la mwisho kwa a) ni: 1.2552 b) Tathmini _10 sqrt(0.6) Step 1: Andika upya mzizi wa mraba kama nguvu na 0.6 kama sehemu. _10 sqrt(0.6) = _10 (0.6)^1/2 = _10 ((6)/(10))^1/2 = _10 ((3)/(5))^1/2 Step 2: Tumia sheria ya nguvu ya logariti (_b (M^p) = p _b M). _10 ((3)/(5))^1/2 = (1)/(2) _10 ((3)/(5)) Step 3: Tumia sheria ya logariti ya kugawanya (_b ((M)/(N)) = _b M - _b N). (1)/(2) _10 ((3)/(5)) = (1)/(2) (_10 3 - _10 5) Step 4: Eleza _10 5 kwa kutumia _10 10 na _10 2. Kumbuka _10 10 = 1. _10 5 = _10 ((10)/(2)) = _10 10 - _10 2 = 1 - _10 2 Step 5: Badilisha usemi wa _10 5 kwenye mlinganyo wa awali. (1)/(2) (_10 3 - (1 - _10 2)) = (1)/(2) (_10 3 - 1 + _10 2) Step 6: Badilisha thamani ulizopewa. (1)/(2) (0.4771 - 1 + 0.3010) Step 7: Fanya hesabu. (1)/(2) (0.7781 - 1) = (1)/(2) (-0.2219) = -0.11095 Jibu la mwisho kwa b) ni: -0.1110 4 done, 1 left today. You're doing great. Tuma swali linalofuata 📸