Question

50 PTS URGENTE LOGARITMO

Answer 1

 0.04 = 4 / 100

 log(3) = 0.47712

 log(5) = 0.69897  

 1/3 log3/5^4( 0.04 )

 na base 10

 1/3 * log(4/100) / log(3/5^4)

 duas propriedades dos logaritmos  

 log(a/b) / log(c/d) = log(b/a) / log(d/c)

 log(a^n) = nlog(a)

 log(4/100) / log(3/5^4) =

 log(100/4) / log(5^4/3) =

 log(25) / log(5^4/3) =  

 log(5^2) / log(5^4/3) =

 2log(5)/(4log(5) - log(3)) =  

 2*0.69897 / (4*0.69897 - 0.47712) = 0.60288

 1/3 log3/5^4( 0.04 ) = 0.60288/3 = 0.20096

Answer 2

$\mathsf{\frac{1}{3}\log_{\frac{3}{5^4}} 0.04 = \dfrac{1}{3}\cdot \dfrac{\log0.04}{\log\frac{3}{5^4}} =} \\ \mathsf{\dfrac{1}{3}\cdot \dfrac{\log\frac{1}{25}}{\log3 - 4\log5} = \dfrac{1}{3}\cdot \dfrac{2\log\frac{1}{5}}{\log 3 - 4\log5} = } \\ \mathsf{\dfrac{1}{3} \cdot \dfrac{2(\log 1 - \log5)}{\log3-4\log5} = \dfrac{1}{3} \cdot \dfrac{2\log5}{4\log5 - \log3}}$

Sabemos que log 3 é aproximadamente 0,47712 e log 5 é aproximadamente 0,69897.

Assim:

$\mathsf{\dfrac{1}{3} \cdot \dfrac{2\log 5}{4\log5- \log 3} = \dfrac{1}{3}\cdot \dfrac{2\cdot0,69897}{4\cdot0,69897 - 0.47712} = 0.20096}$