Question

Calcule os seguintes logaritmos

a) log16 3$\sqrt{8}$
b) log4 √2
c) log 7/3 9/49
d) log 1,25 0,64

Answer 1

João

Ajudo com as três primeiras
Na sequência alfabética, chamando y, z,k aos logaritmos correspndentes

$(16)^y = \sqrt[3]{8} \\ \\ (2^4)^y=(2^3)^ \frac{1}{3} \\ \\ 4y=1 \\ \\ y= \frac{1}{4} \\ \\ \\ 4^z= 2^{ \frac{1}{2} } \\ \\ (2^2)^z= 2^{ \frac{1}{2} } \\ \\ 2z= \frac{1}{2} \\ \\ z= \frac{1}{4} \\ \\ \\ ( \frac{7}{3}) ^{k} = \frac{9}{49} \\ \\ (\frac{7}{3})^k=( \frac{3}{7})^2 \\ \\ ( \frac{7}{3})^k = (\frac{7}{3}) ^{-2} \\ \\ k=-2$

Answer 2

Oie, Td Bom?!

a)

$= log_{16}( \sqrt[3]{8} )$

$= log_{2 {}^{4} }( \sqrt[3]{2 {}^{3} } )$

$= log_{2 {}^{4} }(2)$

$= \frac{1}{4} \: . \: log_{2}(2)$

$= \frac{1}{4} \: . \: 1$

$= \frac{1}{4}$

b)

$= log_{4}( \sqrt{2} )$

$= log_{2 {}^{2} }(2 {}^{ \frac{1}{2} } )$

$= \frac{ \frac{1}{2} }{2} \: . \: log_{2}(2)$

$= \frac{ \frac{1}{2} }{2} \: . \: 1$

$= \frac{ \frac{1}{2} }{2}$

$= \frac{1}{2} \div 2$

$= \frac{1}{2} \: . \: \frac{1}{2}$

$= \frac{1}{4}$

c)

$= log_{ \frac{7}{3} }( \frac{9}{49} )$

$= log_{ \frac{7}{3} }(( \frac{3}{7}) {}^{2} )$

$= log_{ \frac{7}{3} }((( \frac{3}{7} ) {}^{1}) {}^{2} )$

$= log_{ \frac{7}{3} }((( \frac{7}{3} ) {}^{ - 1} ) {}^{2} )$

$= log_{ \frac{7}{3} }(( \frac{7}{3} ) {}^{ - 2} )$

$= - 2 log_{ \frac{7}{3} }( \frac{7}{3} )$

$= - 2 \: . \: 1$

$= - 2$

d)

$= log_{1,25}(0,64)$

$= log_{ \frac{5}{4} }( \frac{16}{25} )$

$= log_{ \frac{5}{4} }(( \frac{4}{5} ) {}^{2} )$

$= log_{ \frac{5}{4} }((( \frac{4}{5}) {}^{1}) {}^{2} )$

$= log_{ \frac{5}{4} }(( \frac{5}{4}) {}^{ - 1}) {}^{2} )$

$= log_{ \frac{5}{4} }(( \frac{5}{4}) {}^{ - 2} )$

$= - 2 log_{ \frac{5}{4} }( \frac{5}{4} )$

$= - 2 \: . \: 1$

$= - 2$

Att. Makaveli1996