Теория:

Если $a>0,\phantom{\rule{0.147em}{0ex}}a\ne 1,\phantom{\rule{0.147em}{0ex}}b>0,\phantom{\rule{0.147em}{0ex}}c>0,\phantom{\rule{0.147em}{0ex}}c\ne 1,\phantom{\rule{0.147em}{0ex}}$ то верно равенство
${\mathit{log}}_{a}b=\frac{{\mathit{log}}_{c}b}{{\mathit{log}}_{c}a}$
Пример:
1.${\mathit{log}}_{2}3=\frac{\mathit{lg}\phantom{\rule{0.147em}{0ex}}3}{\mathit{lg}\phantom{\rule{0.147em}{0ex}}2}$

2.${\mathit{log}}_{3}2=\frac{{\mathit{log}}_{7}2}{{\mathit{log}}_{7}3}$

Если $a>0,\phantom{\rule{0.147em}{0ex}}a\ne 1,\phantom{\rule{0.147em}{0ex}}b>0,\phantom{\rule{0.147em}{0ex}}b\ne 1,$ то верно равенство ${\mathit{log}}_{a}b=\frac{1}{{\mathit{log}}_{b}a}$
Пример:
${\mathit{log}}_{7}2=\frac{1}{{\mathit{log}}_{2}7}$
Если $a>0,\phantom{\rule{0.147em}{0ex}}a\ne 1,\phantom{\rule{0.147em}{0ex}}b>0,\phantom{\rule{0.147em}{0ex}}r\ne 0,$ то верно равенство ${\mathit{log}}_{a}b={\mathit{log}}_{{a}^{r}}{b}^{r}$
Пример:
1.${\mathit{log}}_{5}3={\mathit{log}}_{{5}^{2}}{3}^{2}$

2.${\mathit{log}}_{3}2={\mathit{log}}_{{3}^{-1}}{2}^{-1}$

3.${\mathit{log}}_{7}11={\mathit{log}}_{\sqrt{7}}\sqrt{11}$