Without using a mathematical table, solve for x: log base 2 (3x−4) = 1/3 log base 2 (8x6 - 2)

Question

ScanSolve AI Tutor · Accepted Answer

Hapa kuna hatua za kutatua mlinganyo wa logariti:

Mlinganyo uliotolewa ni:

\log_2 (3x - 4) = \frac{1}{3} \log_2 (8x^6) - 2

Rahisisha upande wa kulia wa mlinganyo.
Tumia sheria ya logariti $\log_b (MN) = \log_b M + \log_b N$ :
$\frac{1}{3} \log_2 (8x^6) - 2 = \frac{1}{3} (\log_2 8 + \log_2 x^6) - 2$
Tunajua kuwa $\log_2 8 = \log_2 (2^3) = 3$ :
$= \frac{1}{3} (3 + \log_2 x^6) - 2$
Gawanya $\frac{1}{3}$ ndani ya mabano:
$= 1 + \frac{1}{3} \log_2 x^6 - 2$
Unganisha namba zisizobadilika:
$= \frac{1}{3} \log_2 x^6 - 1$
Tumia sheria ya logariti $\log_b A^n = n \log_b A$ :
$= \frac{1}{3} (6 \log_2 x) - 1$ $= 2 \log_2 x - 1$
Andika upya mlinganyo kwa kutumia upande wa kulia uliorahisishwa.
$\log_2 (3x - 4) = 2 \log_2 x - 1$
Hamisha namba isiyobadilika (-1) upande wa kushoto na uibadilishe kuwa logariti.
$\log_2 (3x - 4) + 1 = 2 \log_2 x$
Kwa kuwa $1 = \log_2 2$ :
$\log_2 (3x - 4) + \log_2 2 = 2 \log_2 x$
Tumia sheria za logariti kwa pande zote mbili.
Upande wa kushoto: Tumia $\log_b A + \log_b B = \log_b (AB)$ .
$\log_2 (2(3x - 4)) = \log_2 (6x - 8)$
Upande wa kulia: Tumia $n \log_b A = \log_b A^n$ .
$2 \log_2 x = \log_2 x^2$
Linganisha hoja (arguments) za logariti kwa kuwa base zao ni sawa.
$\log_2 (6x - 8) = \log_2 x^2$ $6x - 8 = x^2$
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:
$x = 2 \quad \text{au} \quad x = 4$
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.

Without using a mathematical table, solve for x: log base 2 (3x−4) = 1/3 log base 2 (8x6 - 2)

Answer

Step-by-step solution

Rahisisha upande wa kulia wa mlinganyo.

Andika upya mlinganyo kwa kutumia upande wa kulia uliorahisishwa.

Hamisha namba isiyobadilika (-1) upande wa kushoto na uibadilishe kuwa logariti.

Tumia sheria za logariti kwa pande zote mbili.

Linganisha hoja (arguments) za logariti kwa kuwa base zao ni sawa.

Panga upya mlinganyo kuwa mlinganyo wa quadratic na utatue kwa $x$ .

Hakikisha suluhisho ni halali kwa kuangalia masharti ya logariti (hoja lazima ziwe chanya).

Try one yourself

Still stuck on this one?

More Mathematics Questions

Without using a mathematical table, solve for x: log base 2 (3x−4) = 1/3 log base 2 (8x6 - 2)

Answer

Step-by-step solution

Rahisisha upande wa kulia wa mlinganyo.

Andika upya mlinganyo kwa kutumia upande wa kulia uliorahisishwa.

Hamisha namba isiyobadilika (-1) upande wa kushoto na uibadilishe kuwa logariti.

Tumia sheria za logariti kwa pande zote mbili.

Linganisha hoja (arguments) za logariti kwa kuwa base zao ni sawa.

Panga upya mlinganyo kuwa mlinganyo wa quadratic na utatue kwa xxx.

Hakikisha suluhisho ni halali kwa kuangalia masharti ya logariti (hoja lazima ziwe chanya).

Try one yourself

Still stuck on this one?

More Mathematics Questions

Panga upya mlinganyo kuwa mlinganyo wa quadratic na utatue kwa $x$ .