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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A, let's knock this out. Ili kutatua swali hili, tutatumia kanuni ya kubadilisha 'base' ya logariti, ambayo inasema _b a = ( a)/( b), ambapo bila 'base' maalum inamaanisha 'base' 10. Pia tutatumia kanuni za logariti kama vile (xy) = x + y na (x^n) = n x. Tunapewa: 2 = 0.36103 3 = 0.47712 5 = 0.69897 7 = 0.8451 a) _2 (18) Step 1: Badilisha 'base' ya logariti. _2 (18) = ( 18)/( 2) Step 2: Rahisisha 18 kwa kutumia kanuni za logariti. 18 = (2 × 3^2) = 2 + (3^2) = 2 + 2 3 Step 3: Badilisha thamani ulizopewa na ukokotoe. 18 = 0.36103 + 2(0.47712) = 0.36103 + 0.95424 = 1.31527 _2 (18) = (1.31527)/(0.36103) ≈ 3.6431 Step 4: Jibu la mwisho. 3.6431 b) _15 (45) Step 1: Badilisha 'base' ya logariti. _15 (45) = ( 45)/( 15) Step 2: Rahisisha 45 na 15 kwa kutumia kanuni za logariti. 45 = (3^2 × 5) = 2 3 + 5 15 = (3 × 5) = 3 + 5 Step 3: Badilisha thamani ulizopewa na ukokotoe. 45 = 2(0.47712) + 0.69897 = 0.95424 + 0.69897 = 1.65321 15 = 0.47712 + 0.69897 = 1.17609 _15 (45) = (1.65321)/(1.17609) ≈ 1.4057 Step 4: Jibu la mwisho. 1.4057 c) _7 (200) Step 1: Badilisha 'base' ya logariti. _7 (200) = ( 200)/( 7) Step 2: Rahisisha 200 kwa kutumia kanuni za logariti. 200 = (2^3 × 5^2) = 3 2 + 2 5 Step 3: Badilisha thamani ulizopewa na ukokotoe. 200 = 3(0.36103) + 2(0.69897) = 1.08309 + 1.39794 = 2.48103 _7 (200) = (2.48103)/(0.8451) ≈ 2.9358 Step 4: Jibu la mwisho. 2.9358 d) _3 (27) Step 1: Andika 27 kama 'power' ya 3. 27 = 3^3 Step 2: Tumia kanuni ya logariti _b (b^n) = n. _3 (27) = _3 (3^3) = 3 Step 3: Jibu la mwisho. 3 e) _4 (20) Step 1: Badilisha 'base' ya logariti. _4 (20) = ( 20)/( 4) Step 2: Rahisisha 20 na 4 kwa kutumia kanuni za logariti. 20 = (2^2 × 5) = 2 2 + 5 4 = (2^2) = 2 2 Step 3: Badilisha thamani ulizopewa na ukokotoe. 20 = 2(0.36103) + 0.69897 = 0.72206 + 0.69897 = 1.42103 4 = 2(0.36103) = 0.72206 _4 (20) = (1.42103)/(0.72206) ≈ 1.9679 Step 4: Jibu la mwisho. 1.9679 f) _4 (64) Step 1: Andika 64 kama 'power' ya 4. 64 = 4^3 Step 2: Tumia kanuni ya logariti _b (b^n) = n. _4 (64) = _4 (4^3) = 3 Step 3: Jibu la mwisho. 3 Tuma swali linalofuata 📸