\(\dfrac{f'\left(x\right)}{f\left(x\right)}=\dfrac{1}{x}\Rightarrow ln\left|f\left(x\right)\right|=ln\left|x\right|+C\)
\(\Rightarrow f\left(x\right)=e^{ln\left|x\right|+C}\)
\(f\left(1\right)=e\Rightarrow e=e^{ln1+C}\Rightarrow C=1\)
\(\Rightarrow f\left(x\right)=e^{ln\left|x\right|+1}=e.\left|x\right|\)



