Solve 5raise2x+5raise2x+1=100

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ScanSolve AI Tutor · Accepted Answer

Hili hapa suluhisho la mlinganyo:

Tunapewa mlinganyo:

5^{2x} + 5^{2x+1} = 100

Tumia sheria ya vipeo $a^{m+n} = a^m \cdot a^n$ kurahisisha $5^{2x+1}$ .
$5^{2x} + 5^{2x} \cdot 5^1 = 100$
Toa $5^{2x}$ kama kigezo cha kawaida.
$5^{2x}(1 + 5) = 100$
Rahisisha usemi ndani ya mabano.
$5^{2x}(6) = 100$
Gawanya pande zote mbili kwa 6 ili kutenga $5^{2x}$ .
$5^{2x} = \frac{100}{6}$ $5^{2x} = \frac{50}{3}$
Ili kutatua $x$ , tunahitaji kutumia logariti.
Chukua logariti ya msingi 5 pande zote mbili.
$\log_5(5^{2x}) = \log_5\left(\frac{50}{3}\right)$
Tumia sheria ya logariti $\log_b(b^y) = y$ na $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$ .
$2x = \log_5(50) - \log_5(3)$ $2x = \log_5(25 \cdot 2) - \log_5(3)$ $2x = \log_5(25) + \log_5(2) - \log_5(3)$ $2x = \log_5(5^2) + \log_5(2) - \log_5(3)$ $2x = 2 + \log_5(2) - \log_5(3)$
Gawanya pande zote mbili kwa 2 ili kupata $x$ .
$x = \frac{2 + \log_5(2) - \log_5(3)}{2}$ $x = 1 + \frac{\log_5(2) - \log_5(3)}{2}$
Kama unahitaji jibu la nambari, tunaweza kukadiria maadili ya logariti:
$\log_5(2) \approx 0.430676$
$\log_5(3) \approx 0.682606$
$x \approx 1 + \frac{0.430676 - 0.682606}{2}$ $x \approx 1 + \frac{-0.25193}{2}$ $x \approx 1 - 0.125965$ $x \approx 0.874035$

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