Question Number 65115 by hovea cw last updated on 25/Jul/19
$$\mathrm{x}^{\mathrm{x}} =\mathrm{64} \\ $$$$\mathrm{find}\:\mathrm{x} \\ $$
Answered by mr W last updated on 25/Jul/19
$${x}^{{x}} =\mathrm{64} \\ $$$${x}=\mathrm{64}^{\frac{\mathrm{1}}{{x}}} \\ $$$${x}={e}^{\frac{\mathrm{ln}\:\mathrm{64}}{{x}}} \\ $$$$\mathrm{1}=\frac{\mathrm{1}}{{x}}{e}^{\frac{\mathrm{ln}\:\mathrm{64}}{{x}}} \\ $$$$\mathrm{ln}\:\mathrm{64}=\left(\frac{\mathrm{ln}\:\mathrm{64}}{{x}}\right){e}^{\frac{\mathrm{ln}\:\mathrm{64}}{{x}}} \\ $$$$\Rightarrow\frac{\mathrm{ln}\:\mathrm{64}}{{x}}=\mathbb{W}\left(\mathrm{ln}\:\mathrm{64}\right) \\ $$$$\Rightarrow{x}=\frac{\mathrm{ln}\:\mathrm{64}}{\mathbb{W}\left(\mathrm{ln}\:\mathrm{64}\right)}=\frac{\mathrm{ln}\:\mathrm{64}}{\mathrm{1}.\mathrm{223517}}=\mathrm{3}.\mathrm{399122} \\ $$
Commented by hovea cw last updated on 25/Jul/19
sir I don't know a lot of the Lambert omega function so am kinda stuck at the last few lines plz explain how the function comes about sir ����
Commented by mr W last updated on 25/Jul/19
$${Lambert}\:{W}\:{function}\:{is}\:{the}\:{inverse} \\ $$$${function}\:{from}\:{y}={xe}^{{x}} .\:{i}.{e}. \\ $$$${if}\:{y}={xe}^{{x}} \:{then}\:{x}={W}\left({y}\right). \\ $$$${in}\:{our}\:{example}\:{i}\:{have}\:{formed}\:{the} \\ $$$${original}\:{equation}\:{into} \\ $$$$\left(\frac{\mathrm{ln}\:\mathrm{64}}{{x}}\right){e}^{\left(\frac{\mathrm{ln}\:\mathrm{64}}{{x}}\right)} =\mathrm{ln}\:\mathrm{64} \\ $$$${that}\:{means} \\ $$$$\frac{\mathrm{ln}\:\mathrm{64}}{{x}}={W}\left(\mathrm{ln}\:\mathrm{64}\right) \\ $$$$\Rightarrow{x}=\frac{\mathrm{ln}\:\mathrm{64}}{{W}\left(\mathrm{ln}\:\mathrm{64}\right)} \\ $$
Commented by mr W last updated on 25/Jul/19
$${if}\:{you}\:{can}\:{reform}\:{an}\:{equation}\:{into} \\ $$$${the}\:{form}\:\spadesuit{e}^{\spadesuit} =\clubsuit,\:{then}\:{you}\:{can}\:{apply} \\ $$$${Lambert}\:{W}\:{function}\:{as} \\ $$$$\spadesuit={W}\left(\clubsuit\right) \\ $$$${where}\:\spadesuit\:{is}\:{an}\:{expression}\:{containng}\:{x} \\ $$$${and}\:\clubsuit\:{is}\:{a}\:{constant}. \\ $$
Commented by hovea cw last updated on 25/Jul/19
Thank u sir and how do we approximate the value of w or is it a constant number (*-*)
Commented by mr W last updated on 25/Jul/19
$${in}\:{internet}\:{you}\:{may}\:{find}\:{online} \\ $$$${calculator}\:{for}\:{W}\:{function}\:{values}. \\ $$$${but}\:{since}\:{W}\left(\mathrm{ln}\:\mathrm{64}\right)\:{is}\:{the}\:{root}\:{of} \\ $$$${xe}^{{x}} =\mathrm{ln}\:\mathrm{64} \\ $$$${you}\:{may}\:{get}\:{the}\:{root}\:{via}\:{graphic} \\ $$$${method}.\:{i}\:{use}\:{e}.{g}.\:{an}\:{app}\:{named} \\ $$$${Grapher}. \\ $$