Question

utilizando as propriedades de logaritmos, determine o valor da expressão abaixo:
a) 1
b)2
c)10
d)12​

Answer 1

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☞ b) 2 ✅

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$\green{\rm\underline{EXPLICAC_{\!\!\!,}\tilde{A}O~PASSO{-}A{-}PASSO~~~}}$✍

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☔ Oi, Brendha. Uma importante propriedade da função logarítmica é:

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$\large\red{\boxed{\pink{\boxed{\begin{array}{rcl}&&\\&\orange{\rm log_c\left(\dfrac{a}{b}\right) \iff log_c(a) - log_c(b) }&\\&&\\\end{array}}}}}$

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☔ Com isso temos que:

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$\LARGE\blue{\sf log_2(12) - log_2(3) = log_2(12/3) }$

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$\LARGE\blue{\sf = log_2(4) }$

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☔ Sabemos que:

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$\large\red{\boxed{\pink{\boxed{\begin{array}{rcl}&&\\&\orange{\rm log_b(a) = c \iff b^c = a}&\\&&\\\end{array}}}}}$

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$\LARGE\blue{\sf log_2(4) = x }$

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$\LARGE\blue{\sf 2^x = 4 }$

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$\LARGE\blue{\sf 2^x = 2^2 }$

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$\LARGE\blue{\sf log(2^x) = log(2^2) }$

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☔ Pela Regra do Tombo temos que:

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$\large\red{\boxed{\pink{\boxed{\begin{array}{rcl}&&\\&\orange{\rm log_c(a^b) \iff b \cdot log_c(a)}&\\&&\\\end{array}}}}}$

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$\LARGE\blue{\sf x \cdot log(2) = 2 \cdot log(2) }$

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$\LARGE\blue{\text{ \sf \dfrac{x \cdot log(\diagup\!\!\!\!{2})}{log(\diagup\!\!\!\!{2})} = \dfrac{2 \cdot log(\diagup\!\!\!\!{2})}{log(\diagup\!\!\!\!{2})} }}$

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$\LARGE\blue{\sf x = 2 }$

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$\Huge\green{\boxed{\rm~~~\red{b)}~\blue{2}~~~}}$ ✅

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$\bf\large\red{\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}$☁

☕ $\bf\Large\blue{Bons\ estudos.}$

($\orange{D\acute{u}vidas\ nos\ coment\acute{a}rios}$) ☄

$\bf\large\red{\underline{\qquad \qquad \qquad \qquad \qquad \qquad \quad }}\LaTeX$✍

❄☃ $\sf(\gray{+}~\red{cores}~\blue{com}~\pink{o}~\orange{App}~\green{Brainly})$ ☘☀

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$\gray{"Absque~sudore~et~labore~nullum~opus~perfectum~est."}$