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 hatua za kutatua mlinganyo wa logariti: Mlinganyo uliotolewa ni: _2 (3x - 4) = (1)/(3) _2 (8x^6) - 2 Step 1: Rahisisha upande wa kulia wa mlinganyo. Tumia sheria ya logariti _b (MN) = _b M + _b N: (1)/(3) _2 (8x^6) - 2 = (1)/(3) (_2 8 + _2 x^6) - 2 Tunajua kuwa _2 8 = _2 (2^3) = 3: = (1)/(3) (3 + _2 x^6) - 2 Gawanya (1)/(3) ndani ya mabano: = 1 + (1)/(3) _2 x^6 - 2 Unganisha namba zisizobadilika: = (1)/(3) _2 x^6 - 1 Tumia sheria ya logariti _b A^n = n _b A: = (1)/(3) (6 _2 x) - 1 = 2 _2 x - 1 Step 2: Andika upya mlinganyo kwa kutumia upande wa kulia uliorahisishwa. _2 (3x - 4) = 2 _2 x - 1 Step 3: Hamisha namba isiyobadilika (-1) upande wa kushoto na uibadilishe kuwa logariti. _2 (3x - 4) + 1 = 2 _2 x Kwa kuwa 1 = _2 2: _2 (3x - 4) + _2 2 = 2 _2 x Step 4: Tumia sheria za logariti kwa pande zote mbili. Upande wa kushoto: Tumia _b A + _b B = _b (AB). _2 (2(3x - 4)) = _2 (6x - 8) Upande wa kulia: Tumia n _b A = _b A^n. 2 _2 x = _2 x^2 Step 5: Linganisha hoja (arguments) za logariti kwa kuwa base zao ni sawa. _2 (6x - 8) = _2 x^2 6x - 8 = x^2 Step 6: Panga upya mlinganyo kuwa mlinganyo wa quadratic na utatue kwa x. x^2 - 6x + 8 = 0 Factor mlinganyo wa quadratic: (x - 2)(x - 4) = 0 Hii inatoa suluhisho mbili zinazowezekana: x = 2 au x = 4 Step 7: Hakikisha suluhisho ni halali kwa kuangalia masharti ya logariti (hoja lazima ziwe chanya). Kwa 3x - 4: Kama x = 2: 3(2) - 4 = 6 - 4 = 2 > 0. Halali. Kama x = 4: 3(4) - 4 = 12 - 4 = 8 > 0. Halali. Kwa 8x^6: Kama x = 2: 8(2^6) = 8(64) = 512 > 0. Halali. Kama x = 4: 8(4^6) = 8(4096) = 32768 > 0. Halali. Kwa kuwa masharti yote yametimizwa, suluhisho zote mbili ni halali. Majibu ya mwisho ni: x = 2 au x = 4 3 done, 2 left today. You're making progress.