Question Number 221504 by ajfour last updated on 07/Jun/25
$${Find}\:{x}\:{if}\:\:\:\:\frac{{x}^{\mathrm{4}} }{\mathrm{4}^{{x}} }=\frac{\mathrm{4}^{{x}} }{{x}^{\mathrm{4}} }\:\:.\:\:\:{x}\in\mathbb{R} \\ $$
Answered by mr W last updated on 07/Jun/25
$$\frac{{x}^{\mathrm{4}} }{\mathrm{4}^{{x}} }=\frac{\mathrm{4}^{{x}} }{{x}^{\mathrm{4}} } \\ $$$$\Rightarrow\frac{{x}^{\mathrm{4}} }{\mathrm{4}^{{x}} }=\mathrm{1}\:\:\:\:\:\left(−\mathrm{1}\:{rejected}\right) \\ $$$$\Rightarrow\mathrm{4}^{\frac{{x}}{\mathrm{4}}} =\pm{x} \\ $$$$\Rightarrow{e}^{\frac{{x}\mathrm{ln}\:\mathrm{4}}{\mathrm{4}}} =\pm{x} \\ $$$$\Rightarrow{xe}^{−\frac{{x}\mathrm{ln}\:\mathrm{4}}{\mathrm{4}}} =\pm\mathrm{1} \\ $$$$\Rightarrow\left(−\frac{{x}\mathrm{ln}\:\mathrm{4}}{\mathrm{4}}\right){e}^{−\frac{{x}\mathrm{ln}\:\mathrm{4}}{\mathrm{4}}} =\pm\frac{\mathrm{ln}\:\mathrm{4}}{\mathrm{4}} \\ $$$$\Rightarrow−\frac{{x}\mathrm{ln}\:\mathrm{4}}{\mathrm{4}}={W}\left(\pm\frac{\mathrm{ln}\:\mathrm{4}}{\mathrm{4}}\right) \\ $$$$\Rightarrow{x}=−\frac{\mathrm{4}}{\mathrm{ln}\:\mathrm{4}}{W}\left(\pm\frac{\mathrm{ln}\:\mathrm{4}}{\mathrm{4}}\right)=\begin{cases}{−\mathrm{0}.\mathrm{766664696}}\\{\mathrm{2}}\\{\mathrm{4}}\end{cases} \\ $$
Commented by ajfour last updated on 07/Jun/25
$${thank}\:{you}\:{sir}.\:{Correct}\:{i}\:{had}\:{seen}\:{on}\:{grapher}. \\ $$