Question

1) sendo logb a=4 e logb C =1 encontre o valor de
2,3,4 estão no anexo ​

Answer 1

$\large\boxed{\begin{array}{l}\rm 1)~\sf \ell og_ba=4~~\ell og_bc=1\\\rm a)~\sf\ell og_b(ac)=\ell og_ba+\ell og_bc=4+1=5\\\rm b)~\sf\ell og_b\bigg(\dfrac{a}{b}\bigg)=\ell og_ba-\ell og_bc=4-1=3\\\\\rm c)~\sf \ell og_b(ac)^2=2\ell o_b(ac)=2\cdot5=10\\\rm d)~\sf \ell og_b(\sqrt{a}\cdot c)=\dfrac{1}{2}\ell og_ba+\ell og_bc\\\sf \ell og_b(\sqrt{a}\cdot c)=\dfrac{4}{2}+1=2+1=3\end{array}}$

$\large\boxed{\begin{array}{l}\rm 2)~\sf \ell og_xa=5~~\ell og_xb=2~~\ell og_xc=-1\\\rm a)~\sf \ell og_x(abc)=\ell og_xa+\ell og_xb+\ell og_xc\\\sf \ell og_x(abc)=5+2-1=6\\\rm b)~\sf \ell og_x\dfrac{a^2 b^3}{c^4}=2\ell og_xa+3\ell og_xb-4\ell og_xc\\\\\sf\ell og_x\dfrac{a^2 b^3}{c^4}=2\cdot5+3\cdot2-4\cdot(-1)\\\\\sf \ell og_x\dfrac{a^2b^3}{c^4}=10+6+4=20\end{array}}$

$\large\boxed{\begin{array}{l}\rm 3)~\sf log2=a~~log3=b\\\begin{array}{c|c}\sf180&\sf2\\\sf 90&\sf2\\\sf45&\sf3\\\sf15&\sf3\\\sf5&\sf5\\\sf1\end{array}\\\sf 180=2^2\cdot 3^2\cdot 5\\\sf \ell og180=\ell og(2^2\cdot 3^2\cdot 5)\\\sf \ell og180=2\ell og2+2\ell og3+\ell og5\\\sf \ell og5=\ell og\dfrac{10}{2}=\ell og10-\ell og2=1-a\\\sf \ell og180= 2a+2b+1-a\\\sf \ell og180=a+2b+1\end{array}}$

$\boxed{\begin{array}{l}\rm 4)~\sf \ell og2=a~~\ell og3=b\\\rm a)~\sf \ell og32=\ell og2^5=5\ell og2=5a\\\rm b)~\sf \ell og25=\ell og5^2=2\ell og5=2b\\\rm c)~\sf \ell og\dfrac{81}{\sqrt{3}}=\ell og 81\cdot 3^{-\frac{1}{2}}=\ell og3^4\cdot 3^{-\frac{1}{2}}\\\\\sf \ell og\dfrac{81}{\sqrt{3}}= \ell og3^{\frac{7}{2}}=\dfrac{7}{2}\ell og3=\dfrac{7b}{2}\\\\\rm c)~\sf \ell og(8\cdot \sqrt{27})=\ell og(2^3\cdot 3^{\frac{3}{2}})\\\sf \ell og(8\cdot\sqrt{27})= 3\ell og2+\dfrac{3}{2}\ell og3=3a+\dfrac{3b}{2}\end{array}}$