You may have simplified correctly for the first two steps, but then made a mistake with the factor 4. Start with the power property, then the quotient property: $\displaystyle{\log}_{{{10}}}{{\left(\frac{{100}}{{a}}\right)}}^{{4}}={4}\cdot{\log}_{{10}}{\left(\frac{{100}}{{a}}\right)}={4}{\left({\log}_{{10}}{100}-{\log}_{{10}}{a}\right)}\usepackage{cmbright}\Huge\$

If you have a term such as $\displaystyle{4}\cdot{\log}_{{{10}}}{a}\usepackage{cmbright}\Huge\$ , you may not rewrite it with the factor 4 inside the logarithm.