Question

Calcule o valor de cada logaritmo a seguir:
a) log 1/16 na base 1/2
b) log 1/9 na base 3
c) log 0,1 na base 10
d log 125 na base 1/5

Answer 1

a) Vamos adotar $log2 = 0,30$.

$log_{ \frac{1}{2}} \dfrac{1}{16} = \dfrac{log \frac{1}{16} }{log \frac{1}{2} } = \dfrac{log1-log16}{log1-log2} = \dfrac{log1 - log2^4}{log1-log2} = \dfrac{log1 - 4*log2}{log1-log2}\\\\\\\ \dfrac{log1 - 4*log2}{log1-log2} = \dfrac{0 - 4*0,3}{0-0,3} = \dfrac{0 - 1,2}{0-0,3} = \dfrac{-1,2}{-0,3} = 4$

b) Vamos adotar $log3 = 0,47$.

$log_3 \dfrac{1}{9} = \dfrac{log \frac{1}{9} }{log3} = \dfrac{log1 - log9}{log3} = \dfrac{1-log3^2}{log3} = \dfrac{log1-2*log3}{log3} \\\\\\ \dfrac{log1-2*log3}{log3} = \dfrac{0-2*0,47}{0,47} = \dfrac{-0,94}{0,47} = -2$

c) 

$log_{10}0,1 = log_{10} \dfrac{1}{10} = \dfrac{log \frac{1}{10} }{log10} = \dfrac{log 1-log10}{log10} = \dfrac{0-1}{1} = -1$

d) Vamos adotar $log5 = 0,6$.

$log_{ \frac{1}{5}}125 = \dfrac{log125}{log1-log5} = \dfrac{log5^3}{log1-log5} = \dfrac{3*log5}{log1-log5}\\\\\\\\\dfrac{3*log5}{log1-log5} = \dfrac{3*0,6}{0-0,6} = \dfrac{1,8}{-0,6} = -3$