begin{equation}
    log mu_{ijk}= lambda + lambda_i ^I + lambda_j ^J+lambda_k ^K+lambda_{ij} ^{IJ}+lambda_{ik} ^{IK}+lambda_{jk} ^{JK}+lambda_{ijk} ^{IJK} label{eq:3.1}
    end{equation}

documentclass{article}
usepackage{amsmath}
begin{document}
begin{equation}
    log mu_{ijk}= lambda + lambda_i ^I + lambda_j ^J+lambda_k ^K+lambda_{ij} ^{IJ}+lambda_{ik} ^{IK}+lambda_{jk} ^{JK}+lambda_{ijk} ^{IJK} tag{3.1}label{eq:3.1}
    end{equation}
end{document}

numberwithin{equation}{section}

documentclass{article}
usepackage{amsmath}
numberwithin{equation}{section}

begin{document}

section{Test}
Some text

section{Another Test}
Some text

section{An equation}
begin{equation}
log mu_{ijk}= lambda + lambda_i ^I + lambda_j ^J+lambda_k ^K+lambda_{ij} ^{IJ}+lambda_{ik} ^{IK}+lambda_{jk} ^{JK}+lambda_{ijk} ^{IJK} label{eq:3.1}
end{equation}

end{document}

documentclass{article}
    usepackage{amsmath}
    begin{document}
    begin{equation}tag{3.1}
    begin{split}
        log mu_{ijk}= lambda + lambda_i ^I + lambda_j ^J+lambda_k ^K+lambda_{ij} ^{IJ}+lambda_{ik} ^{IK}+lambda_{jk} ^{JK}+lambda_{ijk} ^{IJK} tag{3.1}label{eq:3.1}

    end{split}
        end{equation}
    end{document}

renewcommand{theequation}{Eq. thesection.arabic{equation}}