Question

se log 2 na base 7=x e log 3 na base 7=y, qual o valor de log 12 na base 18?

Answer 1

Utilizando a lei de mudança de base para logaritmos:

Mudando de uma base $b$ para uma base $c$:

$\mathrm{\ell og}_{b\,}a=\dfrac{\mathrm{\ell og}_{c\,}a}{\mathrm{\ell og}_{c\,}b}$


$x=\mathrm{\ell og}_{7\,}2\\ \\ y=\mathrm{\ell og}_{7\,}3\\ \\ \\ \mathrm{\ell og}_{7\,}12=\mathrm{\ell og}_{7}\left(2^{2}\cdot 3 \right )\\ \\ \mathrm{\ell og}_{7\,}12=\mathrm{\ell og}_{7}\left(2^{2} \right )+\mathrm{\ell og}_{7\,}3\\ \\ \mathrm{\ell og}_{7\,}12=2\mathrm{\,\ell og}_{7\,}2+\mathrm{\ell og}_{7\,}3\\ \\ \boxed{\mathrm{\ell og}_{7\,}12=2x+y}\\ \\ \\ \mathrm{\ell og}_{7\,}18=\mathrm{\ell og}_{7}\left(2 \cdot 3^{2} \right )\\ \\ \mathrm{\ell og}_{7\,}18=\mathrm{\ell og}_{7\,}2+\mathrm{\ell og}_{7}\left(3^{2} \right )\\ \\ \mathrm{\ell og}_{7\,}18=\mathrm{\ell og}_{7\,}2+2\mathrm{\,\ell og}_{7\,}3\\ \\ \boxed{\mathrm{\ell og}_{7\,}18=x+2y}$


Utilizando a lei de mudança de base para

$a=12,\;\;\;b=18,\;\;\;c=7$

temos

$\mathrm{\ell og}_{18\,}12=\dfrac{\mathrm{\ell og}_{7\,}12}{\mathrm{\ell og}_{7\,}18}\\ \\ \boxed{\mathrm{\ell og}_{18\,}12=\dfrac{2x+y}{x+2y}}$