Putem folosi proprietatea urmatoare a logaritmului:
[tex]log_ax+log_ay=log_a(xy)[/tex]
Vom avea nevoie si de urmatoarea proprietate:
[tex]a^{log_ax}=x[/tex]
[tex]\frac{log_23+log_23^2+...+log_23^{100}}{log_23}+2^{2log_27}=\frac{log_2(3\cdot3^2\cdot...\cdot3^{100})}{log_23}+2^{log_27^2}=\\\\
=\frac{log_2(3^{1+2+...+100})}{log_23}+2^{log_249}=\frac{log_2(3^{\frac{100\cdot101}{2}})}{log_23}+49=\\\\\=frac{log_2(3^{5050})}{log_23}+49=\frac{5050log_23}{log_23}+49=\\\\
=5005+49=5099[/tex]
