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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wacha tutatue swali hili. Tunapewa mlinganyo: _3 p + 3 _p 3 = 4 Step 1: Badilisha msingi wa logariti ya pili. Tumia kanuni ya kubadilisha msingi wa logariti: _b a = (1)/(_a b). Hivyo, _p 3 = (1)/(_3 p). Badilisha hii kwenye mlinganyo asili: _3 p + 3 ((1)/(_3 p)) = 4 _3 p + (3)/(_3 p) = 4 Step 2: Weka x = _3 p ili kurahisisha mlinganyo. Mlinganyo unakuwa: x + (3)/(x) = 4 Step 3: Tatua mlinganyo wa quadratic kwa x. Zidisha pande zote mbili kwa x (kwa kuwa x ≠ 0): x^2 + 3 = 4x Panga upya mlinganyo kuwa fomu ya quadratic: x^2 - 4x + 3 = 0 Faktorisha mlinganyo wa quadratic: (x - 1)(x - 3) = 0 Hii inatoa suluhisho mbili zinazowezekana kwa x: x = 1 au x = 3 Step 4: Badilisha x kurudi kwenye _3 p na utatue kwa p. Kesi 1: x = 1 _3 p = 1 Kutoka kwa ufafanuzi wa logariti (b^c = a ikiwa _b a = c): p = 3^1 p = 3 Kesi 2: x = 3 _3 p = 3 Kutoka kwa ufafanuzi wa logariti: p = 3^3 p = 27 Step 5: Thibitisha suluhisho. Kwa logariti _b a, msingi b lazima uwe b > 0 na b ≠ 1, na hoja a lazima iwe a > 0. Katika mlinganyo wetu, p ni hoja ya _3 p na msingi wa _p 3. Hivyo, p lazima iwe p > 0 na p ≠ 1. Maadili yote mawili p=3 na p=27 yanakidhi masharti haya. Kwa p=3: _3 3 + 3 _3 3 = 1 + 3(1) = 4. (Sahihi) Kwa p=27: _3 27 + 3 _27 3 = 3 + 3((1)/(3)) = 3 + 1 = 4. (Sahihi) Maadili ya p ni: p = 3 au p = 27 That's 4 down, 1 left today. What's next?