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.
ScanSolve AI Answer
Haya, wacha tutatue swali la 4(a). Tunapewa mlinganyo: _m 9 + _(m^2) 3 = (5)/(2) Step 1: Badilisha msingi wa logariti ya pili. Tumia kanuni ya kubadilisha msingi wa logariti: _b a = (_c a)/(_c b). Tutatumia c=m. _(m^2) 3 = (_m 3)/(_m (m^2)) Tunajua kuwa _m (m^2) = 2 _m m = 2 × 1 = 2. Hivyo, _(m^2) 3 = (_m 3)/(2) Mlinganyo wetu unakuwa: _m 9 + (_m 3)/(2) = (5)/(2) Step 2: Rahisisha _m 9. Tunajua 9 = 3^2. Tumia kanuni _b (a^n) = n _b a. _m 9 = _m (3^2) = 2 _m 3 Badilisha hii kwenye mlinganyo: 2 _m 3 + (_m 3)/(2) = (5)/(2) Step 3: Tatua kwa _m 3. Wacha X = _m 3. Mlinganyo unakuwa: 2X + (X)/(2) = (5)/(2) Ili kuondoa sehemu, zidisha pande zote mbili kwa 2: 2(2X) + 2((X)/(2)) = 2((5)/(2)) 4X + X = 5 5X = 5 X = 1 Hivyo, _m 3 = 1. Step 4: Tatua kwa m. Kutoka kwa ufafanuzi wa logariti, ikiwa _b a = c, basi b^c = a. Hivyo, m^1 = 3. m = 3 Kwa kuwa m=3 inakidhi masharti ya msingi wa logariti (m>0 na m ≠ 1), suluhisho ni halali. Thamani ya m ni: m = 3 3 done, 2 left today. You're making progress.