Question

Resolva

$\log_5(0,000064)= \\ \\ \log_{49} \sqrt[3]{7} =$

Answer 1

a)
$\log_5(0,000064)=\log_5\left(\dfrac{64}{1000000}\right)=\log_5\left(\dfrac{2^6}{10^6}\right)\\\\ \log_5(0,000064)=\log_5\left(\dfrac{2^6}{(2\cdot5)^6}\right)=\log_5\left(\dfrac{2^6}{2^6\cdot5^6}\right)\\\\ \log_5(0,000064)=\log_5\left(\dfrac{1}{5^6}\right)\\\\ \log_5(0,000064)=\log_5(5^{-6})~~~\overline{\underline{\text{Propriedade: }\log_b(a^N)=N\cdot\log_b a}}\\\\ \log_5(0,000064)=-6\cdot \log_5(5)\\\\ \log_5(0,000064)=-6\cdot1\\\\ \boxed{\log_5(0,000064)=-6}$


b)
$\log_{49}(\sqrt[3]{7})=\log_{49}(7^{\frac{1}{3}})~~~\overline{\underline{\text{Propriedade: }\log_b(a^N)=N\cdot\log_b a}}\\\\ \log_{49}(\sqrt[3]{7})=\dfrac{1}{3}\cdot\log_{49}(7)\\\\ \log_{49}(\sqrt[3]{7})=\dfrac{1}{3}\cdot\log_{(7^2)}(7)~~~\overline{\underline{\text{Propriedade: }\log_{(b^N)}(a)=\dfrac{1}{N}\cdot\log_b a}} \\\\ \log_{49}(\sqrt[3]{7})=\dfrac{1}{3}\cdot\dfrac{1}{2}\cdot\log_7(7)\\\\ \log_{49}(\sqrt[3]{7})=\dfrac{1}{3}\cdot\dfrac{1}{2}\cdot1\\\\ \boxed{\log_{49}(\sqrt[3]{7})=\dfrac{1}{6}}$