\(\displaystyle{X}\ {\log{{X}}}={4}\) or \(\displaystyle{{\log{{X}}}^{{X}}=}{4}{\quad\text{and}\quad}{X}^{{X}}={b}^{{4}}\), where b is the base of the log. So \(X=4\ if\ b=4.\)

When \(b=10\), \(\displaystyle{X}^{{X}}={10000}.\)

Substitute \(X=5\) and we get \(\displaystyle{5}^{{5}}={3125}\), put \(X=6\) and we get \(\displaystyle{6}^{{6}}={46656}\). The solution is between 5 and 6.

The log of a number is much smaller than the number itself. Write \(\displaystyle{X}=\frac{{4}}{{\log{{X}}}}\).

Substitute \(X=5\), so \(\displaystyle{\log{{5}}}={0.6990}\) and the next estimate of \(\displaystyle{X}=\frac{{4}}{{0.6990}}={5.722}.\)

\(\log 5.722=0.7575\) and the next estimate is \(\displaystyle{X}=\frac{{4}}{{0.7575}}={5.281}.\)

If we continue with these iterations we arrive at \(X=5.4385826959\). This takes only a minute or two on a calculator.