Question

Ajuda galera, urgente.

Calcule o valor da expressão:

$log_{3}$ 5.$log_{25}$ 81

Answer 1

$\mathrm{\ell og}_{3\,}5 \cdot \mathrm{\ell og}_{25\,}81\\ \\ =\mathrm{\ell og}_{3\,}5 \cdot \mathrm{\ell og}_{25}\left(3^{4} \right )\\ \\ =\mathrm{\ell og}_{3\,}5 \cdot 4\mathrm{\,\ell og}_{25\,}3\\ \\ =4 \cdot \left(\mathrm{\ell og}_{3\,}5 \cdot \mathrm{\ell og}_{25\,}3 \right )\\ \\ =4 \cdot \left(\mathrm{\ell og}_{3\,}5 \cdot \dfrac{1}{\mathrm{\ell og}_{3\,}25} \right )\\ \\ =4 \cdot \left(\dfrac{\mathrm{\ell og}_{3\,}5}{\mathrm{\ell og}_{3\,}25} \right )\\ \\ =4 \cdot \left(\dfrac{\mathrm{\ell og}_{3\,}5}{\mathrm{\ell og}_{3}\left(5 \right )^{2}} \right )\\ \\ =4 \cdot \left(\dfrac{\mathrm{\ell og}_{3\,}5}{2\mathrm{\,\ell og}_{3\,}5} \right )\\ \\ =4 \cdot \dfrac{1}{2}\\ \\ =2\\ \\ \\ \boxed{\mathrm{\ell og}_{3\,}5 \cdot \mathrm{\ell og}_{25\,}81=2}$