Solve for x: log 2 (x2 - 9) = 3log 2 2 + 1

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Ili kutatua x, tutatumia kanuni za logariti. Step 1: Rahisisha upande wa kulia wa mlinganyo. Tunajua kuwa log_b b = 1. Kwa hiyo, log₂ 2 = 1. log₂(x² − 9) = 3log₂ 2 + 1; log₂(x² − 9) = 3(1) + 1; log₂(x² − 9) = 3 + 1; log₂(x² − 9) = 4 Step 2: Badilisha mlinganyo wa logariti kuwa mlinganyo wa kielelezo. Kama log_b A = C, basi b^C = A. 2⁴ = x² − 9 Step 3: Kokotoa 2⁴. 16 = x² − 9 Step 4: Tatua mlinganyo wa aljebra kwa x. Ongeza 9 pande zote mbili. 16 + 9 = x²; 25 = x² Step 5: Chukua mzizi wa mraba wa pande zote mbili. x = ±√(25); x = ± 5 Step 6: Thibitisha suluhisho. Kwa logariti, hoja (x² − 9) lazima iwe chanya. Kwa x=5: 5² − 9 = 25 − 9 = 16 > 0. Hii ni halali. Kwa x=−5: (−5)² − 9 = 25 − 9 = 16 > 0. Hii pia ni halali. Jibu la mwisho ni: x = ± 5

Solve for x: log 2 (x2 − 9) = 3log 2 2 + 1

Answer

Step-by-step solution

Rahisisha upande wa kulia wa mlinganyo.

Badilisha mlinganyo wa logariti kuwa mlinganyo wa kielelezo.

Kokotoa $2^4$ .

Tatua mlinganyo wa aljebra kwa $x$ .

Chukua mzizi wa mraba wa pande zote mbili.

Thibitisha suluhisho.

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