[tex]f'(x)=\dfrac2x[/tex]
[tex]\implies f(x)=\displaystyle\frac2x\,\mathrm dx=2\ln|x|+C[/tex]
[tex]f(\sqrt e)=5\implies 5=2\ln|\sqrt e|+C[/tex]
[tex]\implies 5=\dfrac22\ln e+C[/tex]
[tex]\implies C=4[/tex]
[tex]\implies f(x)=2\ln|x|+4[/tex]
[tex]\implies f(e)=2\ln|e|+4=6[/tex]
