Question

Na expressão y= log 8 na base 4+ log 3 na base 9 - log 100 - log0,1, o valor de y é: a) -1 b)-2 c)2 d)0 e)1​

Answer 1

$log_4(8) + log_9(3)-log_{10}(100)-log_{10}0,1 \\ log_{2^2}(2^3)+log_{3^2}(3)-log_{10}(10^2)-log_{10}(\frac{1}{10}) \\ \frac{3}{2}log_{2}(2)+\frac{1}{2}log_{3}(3)-2-log_{10}(10^{-1})$

Só lembrando algumas propriedades que foram usadas:

$log_{a^y}(a^x) = \boxed{\frac{x}{y}log_{a}(a)} \\ log_{a}(a^x) = \boxed{x} \\ log_{a}(a^{y}) = \boxed{y*log_a(a)} \\ log_{a}(a) = 1$

Continuando:

$\frac{3}{2}log_{2}(2)+\frac{1}{2}log_{3}(3)-2-log_{10}(10^{-1}) \\ \frac{3}{2}*1+\frac{1}{2}*1-2-(-1log_{10}(10)) \\ \frac{3}{2}+\frac{1}{2}-2-(-1*1) \\ \frac{3}{2}+\frac{1}{2}-2+1 \\ \frac{3}{2}+\frac{1}{2}-1 = \frac{3+1-2}{2} = \frac{2}{2} = \boxed{1}$

Alternativa E).

Answer 2

log 8 na base 4 + log 3 na base 9 - (log 100) - (log0,1)

log2³ (2²) + log 3 (3²) - (log 100) - (log1/10)

(3log2)/2 + (log3)/2 - (log 100) - (log1/10)

Retirando o log de 1/10

log1/10 = log 1 - log 10

log 1 = 0

log 10 = 1

0 - 1 = - 1

log 0,1 (10) = -1

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3.1/2 + 1/2 - 2 -(- 1)  

3/2 + 1/2 -2 + 1

3/2 + 1/2 - 1  

MMC (2,1) = 2

Divide pelo debaixo e multiplica pelo de cima

(2/2).3 = 3

(2/2).1 = 1  

(2/1).-1 = - 2  

Remontando

3/2 + 1/2 - 2/2 = 2/2

2/2 = 1  

LETRA E