Question

QUAL É O VALOR DE CADA UMA DAS EXPRESSOES SEGUINTE? (A FOTO VAI ESTAR ABAIXO)

a) ㏒5 + ㏒ 1 - ㏒ 10
5 3

b) ㏒ 4 + ㏒ 1 sobre 4
1 sobre4 4

c) ㏒ 1000 + ㏒ 100 + ㏒ 10 + ㏒ 1

d) 3㏒ 2 + 2㏒ 3
3 2

Answer 1

Vamos calcular cada logaritmo de cada expressão individualmente e, então, calcular as expressões, acompanhe:

a)

$log_{_5}5~=~x\\\\\\5~=~5^x\\\\\\5^1~=~5^x\\\\\\\boxed{x~=~1}\\\\\\\\log_{_3}1~=~y\\\\\\1~=~3^y\\\\\\3^0~=~3^y\\\\\\\boxed{y~=~0}\\\\\\\\log10~=~z\\\\\\10~=~10^z\\\\\\10^1~=~10^z\\\\\\\boxed{z~=~1}$

Calculando a expressão:

$log_{_5}5~+~log_{_3}1~-~log10~=~x~+~y~-~z\\\\\\log_{_5}5~+~log_{_3}1~-~log10~=~1~+~0~-~1\\\\\\\boxed{log_{_5}5~+~log_{_3}1~-~log10~=~0}$

b)

$log_{_{\frac{1}{4}}}4~=~x\\\\\\4~=~\left(\frac{1}{4}\right)^{x}\\\\\\4^1~=~\left(4^{-1}\right)^x\\\\\\4^1~=~4^{-x}\\\\\\\boxed{x~=~-1}\\\\\\\\log_{_4}\frac{1}{4}~=~y\\\\\\\frac{1}{4}~=~4^y\\\\\\4^{-1}~=~4^y\\\\\\\boxed{y~=~-1}$

Calculando a expressão:

$log_{_{\frac{1}{4}}}4~+~log_{_4}\frac{1}{4}~=~x~+~y\\\\\\log_{_{\frac{1}{4}}}4~+~log_{_4}\frac{1}{4}~=~-1~+~(-1)\\\\\\\boxed{log_{_{\frac{1}{4}}}4~+~log_{_4}\frac{1}{4}~=~-2}$

c)

$log1000~=~x\\\\\\1000~=~10^x\\\\\\10^3~=~10^x\\\\\\\boxed{x~=~3}\\\\\\\\log100~=~y\\\\\\100~=~10^y\\\\\\10^2~=~10^y\\\\\\\boxed{y~=~2}\\\\\\\\log10~=~z\\\\\\10~=~10^z\\\\\\\boxed{z~=~1}\\\\\\\\log1~=~w\\\\\\1~=~10^w\\\\\\10^0~=~10^w\\\\\\\boxed{w~=~0}$

Calculando a expressão:

$log1000~+~log100~+~log10~+~log1~=~x~+~y~+~z~+~z\\\\\\log1000~+~log100~+~log10~+~log1~=~3~+~2~+~1~+~0\\\\\\\boxed{log1000~+~log100~+~log10~+~log1~=~6}$

d)

$3^{log_{_3}2}~=~x\\\\\\3\!\!\!/^{log_{_3\!\!\!/}2}~=~x\\\\\\\boxed{x~=~2}\\\\\\\\2^{log_{_2}3}~=~y\\\\\\2\!\!\!/^{log_{_2\!\!\!/}3}~=~y\\\\\\\boxed{y~=~3}$

Calculando a expressão:

$3^{log_{_3}2}~+~2^{log_{_2}3}~=~x~+~y\\\\\\3^{log_{_3}2}~+~2^{log_{_2}3}~=~2~+~3\\\\\\\boxed{3^{log_{_3}2}~+~2^{log_{_2}3}~=~5}$