n=100;
x=RandomReal[NormalDistribution[100,5],n];

L=(∏_(i=1)^n)((1/(σ(Sqrt[2π])))*Exp[((-(((x[[i]]-μ))^2))/(2*(σ^2)))])

logL=Log[L]

plot=Plot3D[logL,{μ,-10,110},{σ,-10,10},AxesLabel->{"μ","σ"},ColorFunction->"BlueGreenYellow"]

Solve[(∂_μ) logL==0,μ]

Mean[x]

Solve[(∂_σ) (logL/.μ->Mean[x])==0,σ]

plot2=Plot[logL/.μ->Mean[x],{σ,-10,10}]

Plot3D[logL,{μ,Mean[x],Mean[x]+.001},{σ,-10,10}]