$\underset{ }{{\log }_{a}x}=\alpha ,\underset{ }{{\log }_{b}x}=\beta ,\underset{ }{{\log }_{c}x=\gamma }and\underset{ }{{\log }_{d}x}=\delta$ $(x\ne 1)anda,b,c,d>1$ then  $\underset{abcd}{\log x}$ is

$$\begin{gathered} \underset{x}{\log a}=\frac{1}{\alpha },\underset{x}{\log b}=\frac{1}{\beta },\underset{x}{\log c}=\frac{1}{\gamma },\underset{x}{\log d}=\frac{1}{f} \\ \underset{x}{\log }abcd=\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }+\frac{1}{\delta } \\ \underset{abcd}{\log }x=\frac{1}{\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }+\frac{1}{\delta }} \\ For \alpha ,\beta ,\gamma ,\delta \\ \frac{\alpha +\beta +\gamma e\delta }{4}\ge \frac{4}{\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }+\frac{1}{\delta }} \\ \Rightarrow \frac{\alpha +\beta +\gamma -e\delta }{16}\ge \frac{1}{\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }+\frac{1}{\delta }}\Rightarrow \frac{1}{\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }+\frac{1}{\delta }}\le \frac{\alpha +\beta +\gamma e\delta }{16} \\ \underset{abcd}{\log x}\le \frac{\alpha +\beta +\gamma +e\delta }{16} \end{gathered}$$