2-x-3-4-x-48-x-

Question Number 168434 by mathlove last updated on 10/Apr/22

2^{x} \times 3^{\frac{4}{x}} = 48

x = ?

Answered by nurtani last updated on 10/Apr/22

2^{x} \times 3^{\frac{4}{x}} = 16 \times 3

\Leftrightarrow 2^{x} \times 3^{\frac{4}{x}} = 2^{4} \times 3 {\begin{cases} 2^{x} = 2^{4} \Rightarrow x = 4 \\ 3^{\frac{4}{x}} = 3 \Rightarrow \frac{4}{x} = 1 \Rightarrow x = 4 \end{cases}

∴ x = 4

Answered by mr W last updated on 10/Apr/22

x \ln 2 + \frac{4}{x} \ln 3 = \ln 48

x^{2} \ln 2 - x \ln 48 + 4 \ln 3 = 0

x = \frac{\ln 48 \pm \sqrt{{(\ln 48)}^{2} - 16 (\ln 2) (\ln 3)}}{2 \ln 2}

x = \frac{4 \ln 2 + \ln 3 \pm (4 \ln 2 - \ln 3)}{2 \ln 2}

\Rightarrow x = \frac{8 \ln 2}{2 \ln 2} = 4

\Rightarrow x = \frac{\ln 3}{\ln 2} = \log_{2} 3

Answered by cortano1 last updated on 11/Apr/22

\Rightarrow 2^{x} . 3^{\frac{4}{x}} = 2^{4} . 3

\Rightarrow 2^{x - 4} = 3^{1 - \frac{4}{x}}

\Rightarrow x - 4 = (1 - \frac{4}{x}) \log_{2} (3)

\Rightarrow x (x - 4) = (x - 4) \log_{2} (3)

\Rightarrow (x - 4) {x - \log_{2} (3)} = 0

\Rightarrow {\begin{cases} x = 4 \\ x = \log_{2} (3) \end{cases}

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