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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

$$2^x\times 3^{\frac{4}{x}}=16\times 3$$$$\iff 2^x\times 3^{\frac{4}{x}}=2^4\times 3\{\begin{array}{l}{2}^x=2^4\Rightarrow x=4\\ 3^{\frac{4}{x}}=3\Rightarrow \frac{4}{x}=1\Rightarrow x=4\end{array}$$$$\therefore x=4$$

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

$$x\ln 2+\frac{4}{x}\ln 3=\ln 48$$$$x^2\ln 2-x\ln 48+4\ln 3=0$$$$x=\frac{\ln 48\pm \sqrt{{\left(\ln 48\right)}^2-16\left(\ln 2\right)\left(\ln 3\right)}}{2\ln 2}$$$$x=\frac{4\ln 2+\ln 3\pm (4\ln 2-\ln 3)}{2\ln 2}$$$$\Rightarrow x=\frac{8\ln 2}{2\ln 2}=4$$$$\Rightarrow x=\frac{\ln 3}{\ln 2}={\log }_{2}3$$

Answered by cortano1 last updated on 11/Apr/22

$$\Rightarrow 2^x.3^{\frac{4}{x}}=2^4.3$$$$\Rightarrow 2^{x-4}=3^{1-\frac{4}{x}}$$$$\Rightarrow x-4=(1-\frac{4}{x})\log {}_2\left(3\right)$$$$\Rightarrow x(x-4)=(x-4)\log {}_2\left(3\right)$$$$\Rightarrow (x-4)\{x-\log {}_2\left(3\right)\}=0$$$$\Rightarrow \{\begin{array}{l}x=4\\ x=\log {}_2\left(3\right)\end{array}$$

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