Question

Sendo logb a=4 e logb c=1, encontre o valor de:

$logb \: (\sqrt{b} \times c)$

Answer 1

Essas aqui são as propriedades que eu vou usar:

$\log_c{(a\cdot b)}=\log_ca+\log_{c}b$

$\log_{c}{a^b}=b\cdot\log_{c}{a}$

$\displaystyle \log_{a}{a}=1$

$\displaystyle \sqrt[m]{a^n}=a^{\frac{n}{m}}$

Com elas, dá para resolver,

$\displaystyle \log_{b}{(\sqrt{b}\cdot c)}$

$\displaystyle \log_{b}{\sqrt{b}}+\log_{b}{c}$

$\displaystyle \log_{b}{b^{\frac{1}{2}}}+1$

$\displaystyle \frac{1}{2} \cdot \log_{b}{b}+1$

$\displaystyle \frac{1}{2} \cdot 1+1$

$\displaystyle \frac{1+2}{2}$

$\displaystyle \fbox{\frac{3}{2}}$