Question

A) E= log2 3√64- log8 1+log4/3
27/64
B) E= log10 0,001-3 log3 3√3- log4 (log3 81)

C) E=log10 4√1000-3 log3⁴- log4(16)+ log7 7²

Answer 1

a)

$log_{_2}\sqrt[3]{64}~=~x\\\\\\log_{_2}4~=~x\\\\\\4~=~2^x\\\\\\2^2~=~2^x\\\\\\\boxed{x~=~2}\\\\\\\\log_{_8}1~=~y\\\\\\1~=~8^y\\\\\\8^0~=~8^y\\\\\\\boxed{y~=~0}\\\\\\\\log_{_\frac{4}{3}}\frac{27}{64}~=~z\\\\\\\frac{27}{64}~=~\left(\frac{4}{3}\right)^z\\\\\\\frac{3^3}{4^3}~=~\left(\frac{4}{3}\right)^z\\\\\\\left(\frac{3}{4}\right)^3~=~\left(\frac{3}{4}\right)^{-z}\\\\\\-z~=~3\\\\\\\boxed{z~=~-3}$

$E~=~x~-~y~+~z\\\\\\E~=~2~-~0~+~(-3)\\\\\\\boxed{E~=~-1}$

b)

$log_{_{10}}0,001~=~x\\\\\\0,001~=~10^x\\\\\\10^{-4}~=~10^x\\\\\\\boxed{x~=~-4}\\\\\\\\3log_{_3}\sqrt[3]{3}~=~y\\\\\\log_{_3}\sqrt[3]{3}~=~\frac{y}{3}\\\\\\\sqrt[3]{3}~=~3^{\frac{y}{3}}\\\\\\3^{\frac{1}{3}}~=~3^{\frac{y}{3}}\\\\\\\frac{y}{3}~=~\frac{1}{3}\\\\\\\boxed{y~=~1}\\\\\\\\log_{_3}81~=~z\\\\\\81~=~3^z\\\\\\3^4~=~3^z\\\\\\\boxed{z~=~4}\\\\\\\\log_{_4}z~=~w$

$log_{_4}z~=~w\\\\z~=~4^w\\\\\\4~=~4^w\\\\\\\boxed{w~=~1}\\\\\\\\E~=~x~-~y~-~w\\\\\\E~=~-4~-~1~+~1\\\\\\\boxed{E~=~-4}$

c)

$log_{_{10}}\sqrt[4]{1000}~=~x\\\\\sqrt[4]{1000}~=~10^x\\\\\\\sqrt[4]{10^3}~=~10^x\\\\\\10^{\frac{3}{4}}~=~10^x\\\\\\\boxed{x~=~\frac{3}{4}}\\\\\\\\3log_{_3}\,3^4~=~y\\\\\\4~.~3log_{_3}\,3~=~y\\\\\\log_{_3}3~=~\frac{y}{12}\\\\\\3~=~3^{\frac{y}{12}}\\\\\\\frac{y}{12}~=~1\\\\\\\boxed{y~=~12}\\\\\\\\log_{_4}16~=~z\\\\\\16~=~4^z\\\\\\4^2~=~4^z\\\\\\\boxed{z~=~2}$

$log_{_7}7^2~=~w\\\\\\7^2~=~7^w\\\\\\\boxed{w~=~2}\\\\\\\\E~=~x~-~y~-~z~+~w\\\\\\E~=~\frac{3}{4}~-~12~-~2~+~2\\\\\\\boxed{E~=~-\frac{45}{4}}$