Question Number 66543 by hmamarques1994@gmail.com last updated on 17/Aug/19
$$\:\mathrm{3}^{\boldsymbol{{x}}} =\mathrm{3}\boldsymbol{{x}} \\ $$$$\: \\ $$$$\:\boldsymbol{{x}}=? \\ $$
Commented by gunawan last updated on 17/Aug/19
$${x}=\mathrm{1} \\ $$
Answered by mr W last updated on 17/Aug/19
$$\mathrm{3}^{{x}} =\mathrm{3}{x} \\ $$$${e}^{{x}\:\mathrm{ln}\:\mathrm{3}} =\mathrm{3}{x} \\ $$$$\mathrm{3}{xe}^{−{x}\mathrm{ln}\:\mathrm{3}} =\mathrm{1} \\ $$$${xe}^{−{x}\mathrm{ln}\:\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\left(−{x}\mathrm{ln}\:\mathrm{3}\right){e}^{−{x}\mathrm{ln}\:\mathrm{3}} =−\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{3}} \\ $$$$\Rightarrow−{x}\mathrm{ln}\:\mathrm{3}={W}\left(−\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{3}}\right) \\ $$$$\Rightarrow{x}=−\frac{{W}\left(−\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{3}}\right)}{\mathrm{ln}\:\mathrm{3}}=\begin{cases}{−\frac{−\mathrm{1}.\mathrm{098612}}{\mathrm{ln}\:\mathrm{3}}=\mathrm{1}}\\{−\frac{−\mathrm{0}.\mathrm{907473}}{\mathrm{ln}\:\mathrm{3}}=\mathrm{0}.\mathrm{826017}}\end{cases} \\ $$
Commented by Cmr 237 last updated on 17/Aug/19
$$\:{please}\:{sir}\:{i}\:{don}^{.} {t}\:{understand}\:−{your}\:\mathrm{3}^{{rd}} \:{and}\:\mathrm{4}^{{th}} \:{lines}\:{demonstration} \\ $$
Commented by mr W last updated on 17/Aug/19
$${typo}\:{in}\:{third}\:{line}\:{is}\:{fixed}\:{now}. \\ $$
Answered by peter frank last updated on 17/Aug/19
$${use}\:{graph} \\ $$$${y}=\mathrm{3}^{{x}} =\mathrm{3}{x} \\ $$$${y}=\mathrm{3}^{{x}} \\ $$$${y}=\mathrm{3}{x} \\ $$$$ \\ $$