Question

Dada a Expressão logarítmica :

$\maltese~~\mathsf{pr\'incipe~=~\dfrac{1}{2\log_{2} 2016} + \dfrac{1}{ 5\log_{3} 2016} + \dfrac{1}{ 10\log_{7} 2016} }$

Determinar o valor do príncipe

Aí está a diversão ...​

Answer 1

$\maltese ~~ \mathsf{ prince = \frac{1}{ 2 log_{2}2016} + \frac{1}{5 log_{3}2016}+\frac{1}{10 log_{7}2016}}$

Note que

$\mathsf{\frac{1}{2log_{2}2016} =\frac{1}{2} log_{2016}2=\frac{5}{10}log_{2016}2}$

$\mathsf{\frac{1}{5log_{3}2016} = \frac{1}{5} log_{2016}3 = \frac{2}{10} log_{2016}3}$

$\mathsf{\frac{1}{10log_{7}2016} = \dfrac{1}{10} log_{2016}7 }$

Substituindo temos

$\maltese~~ \mathsf{prince = \: \frac{5}{10} log_{2016}2 + \frac{2}{10} log_{2016}3 + \frac{1}{10} log_{2016}(7) }$

$\maltese~~ \mathsf{prince = \: log_{2016} {2}^{ \frac{5}{10} } + log_{2016} {3}^{ \frac{2}{10}}+ log_{2016} {7}^{ \frac{1}{10} } }$

$\maltese~~\mathsf{prince=log_{2016}{({2}^{5}.{3}^{2}.7)}^{\frac{1}{10}}}$

$\large\maltese~~\mathsf{prince=\frac{1}{10} log_{2016}{2016}} = \frac{1}{10}$

Answer 2

Resposta:

$\boxed{\mathtt{1/10}}$

Explicação passo-a-passo:

$\\ \displaystyle \mathsf{Pr\acute{i}ncipe = \frac{1}{2 \cdot \log_2 2016} + \frac{1}{5 \cdot \log_3 2016} + \frac{1}{10 \cdot \log_7 2016}} \\\\\\ \mathsf{Pr\acute{i}ncipe = \frac{1}{2 \cdot \frac{\log 2016}{\log 2}} + \frac{1}{5 \cdot \frac{\log 2016}{\log 3}} + \frac{1}{10 \cdot \frac{\log 2016}{\log 7}}} \\\\\\ \mathsf{Pr\acute{i}ncipe = \frac{\log 2}{2 \cdot \log 2016} + \frac{\log 3}{5 \cdot \log 2016} + \frac{\log 7}{10 \cdot \log 2016}} \\\\\\ \mathsf{Pr\acute{i}ncipe = \frac{5 \cdot \log 2 + 2 \cdot \log 3 + 1 \cdot \log 7}{10 \cdot \log 2016}}$

$\\ \displaystyle \mathsf{Pr\acute{i}ncipe = \frac{\log 2^5 + \log 3^2 + \log 7^1}{10 \cdot \log 2016}} \\\\\\ \mathsf{Pr\acute{i}ncipe = \frac{\log 32 + \log 9 + \log 7}{10 \cdot \log 2016}}$

$\\ \displaystyle \mathsf{Pr\acute{i}ncipe = \frac{\log \left ( 32 \cdot 9 \cdot 7 \right )}{10 \cdot \log 2016}} \\\\\\ \mathsf{Pr\acute{i}ncipe = \frac{\log 2016}{10 \cdot \log 2016}}$

$\displaystyle \boxed{\boxed{\mathsf{Pr\acute{i}ncipe = \frac{1}{10}}}}$