Question Number 222552 by mathlove last updated on 29/Jun/25
$${x}^{{x}} =\mathrm{64}\:\:\:\:\:\:,\:\:{x}=? \\ $$
Answered by mr W last updated on 29/Jun/25
$${x}\mathrm{ln}\:{x}=\mathrm{ln}\:\mathrm{64} \\ $$$$\mathrm{ln}\:{xe}^{\mathrm{ln}\:{x}} =\mathrm{6ln}\:\mathrm{2} \\ $$$$\mathrm{ln}\:{x}={W}\left(\mathrm{6ln}\:\mathrm{2}\right) \\ $$$${x}={e}^{{W}\left(\mathrm{6ln}\:\mathrm{2}\right)} =\frac{\mathrm{6ln}\:\mathrm{2}}{{W}\left(\mathrm{6ln}\:\mathrm{2}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\approx\frac{\mathrm{6ln}\:\mathrm{2}}{\mathrm{1}.\mathrm{223517}}\approx\mathrm{3}.\mathrm{399122}\:\checkmark \\ $$
Commented by mathlove last updated on 30/Jun/25
$${thanks}\:{mr}\:{W} \\ $$
Commented by mathlove last updated on 30/Jun/25
$${thanks}\:{mr}\:{W} \\ $$