How to solve this equation?

Your question is a bit confusing - is this what your mean.

This is wrong (unless your question is wrong) I just realised that I used log base e where yours is log base 10 sorry

You can still do it straight on the web 2 calc.

$$\left[1+\frac{1}{-e^{0.05*log(0.05)}}\right]^{-1}\\\\

\left[1+\frac{1}{-e^{log(0.05^{0.05})}}\right]^{-1}\\\\

\left[1+\frac{1}{-(0.05)^{0.05}}}\right]^{-1}\\\\

\left[\frac{-(0.05)^{0.05}+1}{-(0.05)^{0.05}}\right]^{-1}\\\\

\left[\frac{-(0.05)^{0.05}}{-(0.05)^{0.05}+1}\right]^{+1}\\\\
\frac{-(0.05)^{0.05}}{-(0.05)^{0.05}+1}\\\\$$

Now you can just do it on the calculator - or you could have just done it on the calculator to start with.   

$${\mathtt{\,-\,}}{\frac{{\left({\mathtt{0.05}}\right)}^{{\mathtt{0.05}}}}{\left({\mathtt{1}}{\mathtt{\,-\,}}{\left({\mathtt{0.05}}\right)}^{{\mathtt{0.05}}}\right)}} = -{\mathtt{6.188\: \!641\: \!566\: \!681\: \!630\: \!8}}$$

$$\left({\left({\mathtt{1}}{\mathtt{\,-\,}}{{\left({{\mathtt{e}}}^{\left({\mathtt{0.05}}{\mathtt{\,\times\,}}{ln}{\left({\mathtt{0.05}}\right)}\right)}\right)}}^{-{\mathtt{1}}}\right)}^{-{\mathtt{1}}}\right) = -{\mathtt{6.188\: \!641\: \!566\: \!681\: \!631\: \!1}}$$

NOW FOR A VERSION THAT MAY BE THE REAL VERSION.

$$\left({\left({\mathtt{1}}{\mathtt{\,-\,}}{{\left({{\mathtt{e}}}^{\left({\mathtt{0.05}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{0.05}}\right)\right)}\right)}}^{-{\mathtt{1}}}\right)}^{-{\mathtt{1}}}\right) = -{\mathtt{14.877\: \!856\: \!312\: \!827\: \!660\: \!5}}$$