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Question # Solve for X logX=4

Logarithms
ANSWERED Solve for X $$\displaystyle{\log{{X}}}={4}$$ 2020-11-03

$$\displaystyle{X}\ {\log{{X}}}={4}$$ 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.