Question

Sabendo que log 2 = 0,30 e log 3 = 0,47 determine:
$log_{6}0.75$
​

Answer 1

Para resolver esta expressão logarítmica, vamos precisar utilizar algumas propriedades, acompanhe:

$log_{_6}0,75~=\\\\\\Comecamos~passando~0,75~para~fracao\\\\\\=~log_{_6}\frac{3}{4}\\\\\\Utilizando~a~\underline{propriedade~do~logaritmo~do~quociente}\\\\\\=~log_{_6}3~-~log_{_6}4\\\\\\Podemos~agora~utilizar~a~\underline{propriedade~da~troca~de~base}~para~mudar\\a~base~do~log~de~6~para~10\\\\\\=~\dfrac{log\,3}{log\,6}~-~\dfrac{log\,4}{log\,6}\\\\\\Vamos~agora~fatorar~''6''~e~''4''$

$=~\dfrac{log\,3}{log\,(2\,.\,3)}~-~\dfrac{log\,(2\,.\,2)}{log\,(2\,.\,3)}\\\\\\A~ultima~propriedade~que~vamos~utilizar~\acute{e}~a~do~\underline{logaritmo~do~produto}\\\\\\=~\dfrac{log\,3}{log\,2~+~log\,3}~-~\dfrac{log\,2~+~log\,2}{\,log\,2~+~log\,3}\\\\\\Por~fim,~vamos~\underline{substituir~os~valores}~dados\\\\\\=~\dfrac{0,47}{0,30+0,47}~-~\dfrac{0,30+0,30}{0,47}\\\\\\=~\dfrac{0,47}{0,77}~-~\dfrac{0,60}{0,77}\\\\\\=\,-\dfrac{0,13}{0,77}\\\\\\=~\boxed{-\dfrac{13}{77}}$