Question Number 226958 by efronzo1 last updated on 21/Dec/25
$$\:\:\:\mathrm{3}^{\mathrm{x}} =\mathrm{x}^{\mathrm{9}} \: \\ $$$$\:\:\:\:\mathrm{x}^{\mathrm{2}} =\:..? \\ $$
Answered by TonyCWX last updated on 21/Dec/25
$${x}\mathrm{ln}\left(\mathrm{3}\right)\:=\:\mathrm{9ln}\left({x}\right) \\ $$$${x}^{−\mathrm{1}} \mathrm{ln}\left({x}\right)=\frac{\mathrm{ln}\left(\mathrm{3}\right)}{\mathrm{9}} \\ $$$$\mathrm{ln}\left({x}\right){e}^{−\mathrm{ln}\left({x}\right)} =\frac{\mathrm{ln}\left(\mathrm{3}\right)}{\mathrm{9}} \\ $$$$−\mathrm{ln}\left({x}\right){e}^{−\mathrm{ln}\left({x}\right)} =−\frac{\mathrm{ln}\left(\mathrm{3}\right)}{\mathrm{9}} \\ $$$$−\mathrm{ln}\left({x}\right)={W}\left(−\frac{\mathrm{ln}\left(\mathrm{3}\right)}{\mathrm{9}}\right) \\ $$$${x}={e}^{−{W}\left(−\frac{\mathrm{ln}\left(\mathrm{3}\right)}{\mathrm{9}}\right)} \\ $$
Answered by mr W last updated on 21/Dec/25
$$\mathrm{3}^{{x}} ={x}^{\mathrm{9}} \\ $$$${x}^{\frac{\mathrm{9}}{{x}}} =\mathrm{3} \\ $$$${x}^{\frac{\mathrm{1}}{{x}}} =\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{9}}} =\left(\mathrm{27}\right)^{\frac{\mathrm{1}}{\mathrm{27}}} \\ $$$$\Rightarrow{x}=\mathrm{27}\: \\ $$$${this}\:{is}\:{one}\:{of}\:{the}\:{two}\:{roots}. \\ $$