Question

2vezes logx^4 na base 5 = 10 + log x^3 na base 5.

Answer 1

$\mathbf{Algumas\ das\ propriedades\ dos\ logaritmos:}\\\\ \mathrm{\Rightarrow \log{a^n}=n\log{a}}\\ \mathrm{\Rightarrow \log{a}-\log{b}=\log{\bigg(\dfrac{a}{b}\bigg)}}$

$\textbf{Resolu\c{c}\~ao:}\\\\ \mathrm{2\log_5{x^4}=10+\log_5{x^3}\ \to\ \log_5{(x^4)^2=10+\log_5{x^3}}}\\\\ \mathrm{\log_5{x^8}=10+\log_5{x^3}\ \to\ \log_5{x^8}-\log_5{x^3}=10}\\\\ \mathrm{\log_5{\bigg(\dfrac{x^8}{x^3}\bigg)}=10\ \to\ \log_5{x^5}=10\ \to\ 5^{10}=x^5}\\\\ \mathrm{x^5=(5^2)^5\ \to\ x=\sqrt[5]{(5^2)^5}\ \to\ x=5^2\ \to\ \boxed{\mathbf{x=25}}}$