Question

Sabendo que log 3=a e log 5=b,calcule o valor de
$log_{30}150$


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Answer 1

Primeiro se muda a base do log para 10:

$log_{30}150=log(150)/log(30)$

Depois desmembra o log em múltiplos de 5 e 3:

$=log(5.3.10)/log(3.10)$

$=(log5+log3+log10)/(log3+log10)$

Por fim substitui por a e b:
$=b+a+1/a+1$

Answer 2

Resposta:

$\boxed{\mathtt{\frac{a + b + 1}{a + 1}}}$

Explicação passo-a-passo:

$\\ \displaystyle \mathsf{\log_{30} 150 = \log_{30} \left ( 30 \cdot 5 \right )} \\\\ \mathsf{\qquad \qquad = \log_{30} 30 + \log_{30} 5} \\\\ \mathsf{\qquad \qquad = 1 + \frac{\log 5}{\log 30}} \\\\\\ \mathsf{\qquad \qquad = 1 + \frac{\log 5}{\log (3 \cdot 10)}} \\\\\\ \mathsf{\qquad \qquad = 1 + \frac{\log 5}{\log 3 + \log 10}} \\\\\\ \mathsf{\qquad \qquad = 1 + \frac{b}{a + 1}} \\\\\\ \boxed{\boxed{\mathsf{\log_{30} 150 = \frac{a + 1 + b}{a + 1}}}}$