Question

Sabendo que log 2= a e log 3= b, calcule, em função de a e b :
log 0,024 e log 0,75

Answer 1

$\log(0,024)=\log\left( \dfrac{24}{1.000}\right)\\\\ \log(0,024)=\log(24)-\log(1.000)\\ \log(0,024)=\log(2^3\cdot3)-\log(10^3)\\ \log(0,024)=\log(2)^3+\log(3)-\log_{10}(10)^3\\ \log(0,024)=3\cdot\log(2)+\log(3)-3\cdot\log_{10}(10)\\ \log(0,024)=3\cdot a+b-3\cdot1\\\\ \Large\boxed{\log(0,024)=3a+b-3}$

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$\log(0,75)=\log\left( \dfrac{75}{100}\right)\\\\ \log(0,75)=\log(75)-\log(100)\\ \log(0,75)=\log(5^2\cdot3)-\log(10^2)\\ \log(0,75)=\log(5)^2+\log(3)-\log_{10}(10)^2\\ \log(0,75)=\log\left( \dfrac{10}{2} \right)^2+\log(3)-2\cdot\log_{10}(10)\\ \log(0,75)=[\log(10)-\log(2)]^2+\log(3)+2\cdot\log_{10}(10)\\ \log(0,75)=2\cdot[\log_{10}(10)-\log(2)]+\log(3)+2\cdot\log_{10}(10)\\ \log(0,75)=2\cdot(1-a)+b+2\cdot1\\ \log(0,75)=2-2a+b+2\\\\ \Large\boxed{\log(0,75)=4-2a+b}$